Neutral atom quantum information processor

ABSTRACT

Systems and methods relate to arranging atoms into 1D and/or 2D arrays; exciting the atoms into Rydberg states and evolving the array of atoms, for example, using laser manipulation techniques and high-fidelity laser systems described herein; and observing the resulting final state. In addition, refinements can be made, such as providing high fidelity and coherent control of the assembled array of atoms. Exemplary problems can be solved using the systems and methods for arrangement and control of atoms.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of U.S. application Ser. No.16/630,719, filed on Jan. 13, 2020, which is the U.S. National Stage ofInternational Application No. PCT/US2018/042080, filed on Jul. 13, 2018,published in English, which claims the benefit of priority to U.S.Provisional Application No. 62/531,993, filed on Jul. 13, 2017 and62/589,716, filed on Nov. 22, 2017. The entire teachings of the aboveapplication(s) are incorporated herein by reference.

COPYRIGHT NOTICE

This patent disclosure may contain material that is subject to copyrightprotection. The copyright owner has no objection to the facsimilereproduction by anyone of the patent document or the patent disclosureas it appears in the U.S. Patent and Trademark Office patent file orrecords, but otherwise reserves any and all copyright rights.

TECHNICAL FIELD

This patent relates to quantum computing, and more specifically topreparing and evolving an array of atoms.

SUMMARY OF THE INVENTION

According to some embodiments, the system and methods described hereinrelate to arranging atoms into 1D and/or 2D arrays; exciting the atomsinto Rydberg states and evolving the array of atoms, for example, usinglaser manipulation techniques and high fidelity laser systems describedherein; and observing the resulting final state. In addition,refinements can be made to the system and methods described herein, suchas providing high fidelity and coherent control of the assembled arrayof atoms. Exemplary problems are discussed which can be solved using thesystem and methods described herein.

In one or more embodiments, a method includes forming an array of atomsin a first array state, wherein said forming includes: exciting acrystal with a plurality of discrete adjustable acoustic tonefrequencies, passing a laser through the crystal to create a pluralityof confinement regions, wherein each acoustic tone frequency correspondsto an individual confinement region for a single atom, trapping at leasttwo atoms in at least two of said plurality of confinement regions,correlating the discrete adjustable acoustic tone frequencies toidentify the confinement regions that contain the trapped atoms, andadjusting a spacing between at least two of the trapped atoms bysweeping at least one correlated adjustable acoustic tone frequency;evolving the plurality of atoms in the first array state into aplurality of atoms in a second array state by subjecting at least someof the trapped atoms to photon energy to transition at least some of thetrapped atoms into an excited state; and observing the plurality ofatoms in the second array state.

In one or more embodiments, the excited state is a Rydberg state.

In one or more embodiments, the plurality of atoms in the first arraystate includes between 7 and 51 atoms.

In one or more embodiments, the evolving the plurality of atoms includespreparing at least some of the atoms in the first array state into aZeeman sublevel of the ground state before subjecting at least some ofthe atoms to photon energy.

In one or more embodiments, the preparing the atoms in the first arraystate into a Zeeman sublevel of the ground state includes opticalpumping in a magnetic field.

In one or more embodiments, the subjecting at least some of the atoms tophoton energy includes applying light having two different wavelengths,and wherein the transition of the at least some of the atoms into anexcited state includes a two photon transition.

In one or more embodiments, the two different wavelengths areapproximately 420 nm and approximately 1013 nm.

In one or more embodiments, the method further includes applying a phasegate with a third wavelength.

In one or more embodiments, the third wavelength is approximately 809nm.

In one or more embodiments, the subjecting the at least some of theatoms to photon energy includes applying two half-pi pulses.

In one or more embodiments, the subjecting the at least some of theatoms to photon energy further includes applying a pi pulse between thetwo half-pi pulses.

In one or more embodiments, the trapping the at least two at least twoatoms includes trapping at least two atoms from a cloud of atoms anddispersing atoms from the cloud of atoms not trapped in one of saidplurality of confinement regions.

In one or more embodiments, the crystal and laser comprise a firstcontrol acousto-optic deflector (AOD), and wherein the trapping the atleast two atoms includes trapping atoms from a hold trap array having atleast three traps spaced apart in two dimensions.

In one or more embodiments, the hold trap array is generated by at leastone of at least one hold AOD, a spatial light modulator (SLM), and anoptical lattice.

In one or more embodiments, the method further includes a second controlAOD configured in a crossed relationship with the first control AOD, andwherein: the correlating the discrete adjustable acoustic tonefrequencies to identify the confinement regions that contain the trappedatoms includes correlating with discrete adjustable acoustic tonefrequencies of the first control AOD and the second control AOD, and theadjusting the spacing between the at least two of the trapped atomsincludes sweeping at least one correlated adjustable acoustic tonefrequency of the first control AOD or the second control AOD.

In one or more embodiments, the adjusting the spacing between the atleast two of the trapped atoms further includes adjusting the positionof multiple atoms in a row.

In one or more embodiments, the method further includes: forming asecond array of atoms in a third array state adjacent to the first arrayof atoms, wherein said forming includes: exciting a second crystal witha plurality of second discrete adjustable acoustic tone frequencies,passing a second laser through the second crystal to create a pluralityof second confinement regions, wherein each second acoustic tonefrequency corresponds to an individual second confinement region for asingle atom, trapping at least two second atoms in at least two of saidplurality of second confinement regions, correlating the second discreteadjustable acoustic tone frequencies to identify the second confinementregions that contain the trapped atoms, and adjusting a spacing betweenat least two of the trapped second atoms by sweeping at least one secondcorrelated adjustable acoustic tone frequency; wherein the evolving theplurality of atoms in the first array state into a plurality of atoms ina second array state by subjecting at least some of the trapped atoms tophoton energy to transition the at least some of the trapped atoms intothe excited state further includes evolving the plurality of secondatoms in the third array state into a plurality of second atoms in afourth array state by subjecting at least some of the second trappedatoms to photon energy to transition at least some of the second trappedatoms into an excited state; and wherein the observing the plurality ofatoms in the second array state further includes observing the pluralityof second atoms in the fourth array state.

In one or more embodiments, the adjusting the spacing between at leasttwo of the trapped atoms by sweeping at least one correlated adjustableacoustic tone frequency includes encoding a quantum computing problem;the evolving the plurality of atoms in the first array state into theplurality of atoms in the second array state produces a solution to thequantum computing problem; and the observing the plurality of atoms inthe second array state includes reading out the solution to the quantumcomputing problem.

In one or more embodiments, the quantum computing problem includes atleast one of an Ising-problem and a maximum independent set (MIS)optimization problem.

In one or more embodiments, a system includes: a confinement system forarranging an array of atoms in a first array state, the confinementsystem comprising: a crystal, an adjustable acoustic tone frequencyapplication source configured to selectively apply a plurality ofdiscrete adjustable acoustic tone frequencies to the crystal, and alaser source arranged pass light through the crystal to create aplurality of confinement regions, wherein each acoustic tone frequencycorresponds to an individual confinement region, a source of an atomcloud, the atom cloud capable of being positioned to at least partiallyoverlap with the plurality of confinement regions; an excitation sourcefor evolving at least some of the plurality of atoms in the first arraystate into a plurality of atoms in a second array state, the excitationsource comprising at least one source of photon energy; an observingsystem for observing the plurality of atoms in the second array state.

In one or more embodiments, the excitation source is configured toexcited at least some of the plurality of atoms in the first array stateinto a Rydberg state

In one or more embodiments, the plurality of atoms in the first arraystate includes between 5 and 51 atoms.

In one or more embodiments, the excitation source is configured toexcited at least some of the plurality of atoms in the first array stateinto a Zeeman sublevel of the ground state before subjecting at leastsome of the atoms to photon energy.

In one or more embodiments, claim 0, the excitation source furtherincludes an optical pumping system and a magnetic field generator.

In one or more embodiments, the at least one source of photon energyincludes light sources having a first wavelength and a second wavelengthfor producing a two photon transition of the at least some of theplurality of atoms in the first array state.

In one or more embodiments, the two different wavelengths areapproximately 420 nm and approximately 1013 nm.

In one or more embodiments, the at least one source of photon energyincludes a source having a third wavelength for applying a phase gate.

In one or more embodiments, the third wavelength is approximately 809nm.

In one or more embodiments, the excitation source is configured to applytwo half-pi pulses.

In one or more embodiments, the excitation source is configured to applya pi pulse between the two half-pi pulses.

In one or more embodiments, the confinement system is a first controlacousto-optic deflector (AOD), and wherein the system further includes ahold trap array having at least three traps spaced apart in twodimensions, the hold trap array being generated by a hold trap source.

In one or more embodiments, the hold trap source includes at least oneof at least one hold AOD, a spatial light modulator (SLM), and anoptical lattice.

In one or more embodiments, the system further includes a second controlAOD in a crossed relationship with the first control AOD, wherein thefirst control AOD controls deflection of light beams in a firstdirection, and the second control AOD controls deflection of light beamsfrom the first AOD in a second direction different from the firstdirection.

In one or more embodiments, the confinement system is a first controlacousto-optic deflector (AOD), and wherein the system further includes asecond control AOD in a stacked relationship with the first control AOD,wherein the first control AOD is configured to produce a plurality ofconfinement regions in a first array having a first direction, and thesecond control AOD controls is configured to produce a plurality ofconfinement regions in a second array that is substantially parallel tothe first direction.

In one or more embodiments, a system for controlling an array of trappedatoms includes: a laser source for producing a laser output; a lasersource controller that controls the laser source; a Pound-Drever-Hall(PDH) lock optically coupled to the laser source that receives at leastsome of the laser output and provides a feedback signal to the lasersource controller for stabilizing the laser output; a reference opticalcavity optically coupled to the laser source, the reference opticalcavity configured to receive at least some of the laser output and totransmit a reference optical cavity output, the reference optical cavityoutput corresponding to a portion of the at least some of the laseroutput that falls within a reference optical cavity transmission window;and an optical isolator optically coupled to the reference opticalcavity, the optical isolator configured to split the reference opticalcavity output and to provide at least a portion of the split referenceoptical cavity output to a Fabry-Perot laser diode to injection lock thereference optical cavity output, wherein the optical isolator providesinjection locked light to at least some of the trapped atoms.

In one or more embodiments, the PDH further includes a photodetectorthat receives at least some of the laser output and outputs aphotodetector signal to the laser controller.

In one or more embodiments, the system further includes a second lasersource for providing a second laser output at a different wavelengthfrom the first laser output.

In one or more embodiments, the system further includes: a second lasersource controller that controls the second laser source; a secondPound-Drever-Hall (PDH) lock optically coupled to the second lasersource that receives at least some of the second laser output andprovides a second feedback signal to the second laser source controllerfor stabilizing the second laser output; a second reference opticalcavity optically coupled to the second laser source, the secondreference optical cavity configured to receive at least some of thesecond laser output and to transmit a second reference optical cavityoutput, the second reference optical cavity output corresponding to aportion of the at least some of the second laser output that fallswithin a second reference optical cavity transmission window; and ansecond optical isolator optically coupled to the second referenceoptical cavity, the second optical isolator configured to split thesecond reference optical cavity output and to provide at least a portionof the split second reference optical cavity output to a secondFabry-Perot laser diode to injection lock the second reference opticalcavity output, wherein the second optical isolator provides secondinjection locked light to at least some of the trapped atoms.

In one or more embodiments, the second laser source produces light atapproximately 1013 nm.

In one or more embodiments, the second reference optical cavity and thefirst reference optical cavity are the same element.

In one or more embodiments, the first injection locked light and thesecond injection locked light are provided to the at least some of thetrapped atoms in a counter propagating configuration.

In one or more embodiments, the first laser source produces light atapproximately 420 nm.

In one or more embodiments, the system further includes opticspositioned optically between the optical isolator and the array oftrapped atoms configured to focus the injection locked light onto the atleast some of the trapped atoms.

In one or more embodiments, the system further includes a spatiallyresolved imaging device configured to pick off at least a portion of theinjection locked light in order to align the injection locked light.

These and other capabilities of the disclosed subject matter will bemore fully understood after a review of the following figures, detaileddescription, and claims. It is to be understood that the phraseology andterminology employed herein are for the purpose of description andshould not be regarded as limiting.

BRIEF DESCRIPTION OF FIGURES

Various objectives, features, and advantages of the disclosed subjectmatter can be more fully appreciated with reference to the followingdetailed description of the disclosed subject matter when considered inconnection with the following drawings, in which like reference numeralsidentify like elements.

FIGS. 1A-F show aspects of a system and method for preparing an array ofatoms, according to some embodiments.

FIGS. 2A-B show a phase diagram and buildup of crystalline phases,according to some embodiments.

FIGS. 3A-C show a comparison of the methods described in the presentdisclosure with a simulation, according to some embodiments.

FIGS. 4A-B show scaling behavior, according to some embodiments.

FIGS. 5A-D show arrays of atoms before and after adiabatic evolution,and characteristics thereof, according to some embodiments.

FIGS. 6A-D show graphical representations of oscillations in many-bodydynamics, according to some embodiments.

FIGS. 7A-7D show systems for and graphs characterizing control forsingle-atom placement and manipulation, according to some embodiments.

FIGS. 8A-8C show graphs characterizing single-atom coherence and phasecontrol, according to some embodiments.

FIGS. 9A-9C show graphs characterizing entanglement generation with twoatoms.

FIG. 10 shows a graph representing extension of entangled-state lifetimevia dynamical decoupling, according to an embodiment.

FIGS. 11A-11B show examples of independent sets including maximumindependent sets, according to some embodiments.

FIG. 12 shows an example of a unit disc graph, according to anembodiment.

FIGS. 13A-B show an example of a unit disc graph indicating the maximumindependent set the probability distribution of finding an independentset, according to some embodiments.

FIG. 14 shows a system for two-dimensional ordering of atoms, accordingto an embodiment.

FIGS. 15A-15H show methods for two-dimensional ordering of atoms,according to some embodiments.

FIG. 16 shows a system for two-dimensional ordering of atoms, accordingto an embodiment.

FIGS. 17A-17E show methods for two-dimensional ordering of atoms,according to some embodiments.

FIGS. 18A-18H show methods for two-dimensional ordering of atoms,according to some embodiments.

FIGS. 19A-19B show a system for two-dimensional ordering of atoms,according to an embodiment.

FIGS. 20A-20C show methods for two-dimensional ordering of atoms,according to some embodiments.

FIGS. 21A-21B show methods for two-dimensional ordering of atoms,according to some embodiments.

FIGS. 22A-22P show systems for two-dimensional ordering of atoms,according to some embodiments.

FIG. 23 shows a system for two-dimensional ordering of atoms, accordingto an embodiment.

FIGS. 24A-24E show methods for two-dimensional ordering of atoms,according to some embodiments.

FIGS. 25A-25H show methods for two-dimensional ordering of atoms,according to some embodiments.

FIG. 26 shows an image of an array of traps generated with an SLM,according to an embodiment.

FIG. 27 shows a spatial phase pattern, according to an embodiment.

DETAILED DESCRIPTION

As quantum simulators, fully controlled, coherent many-body quantumsystems can provide unique insights into strongly correlated quantumsystems and the role of quantum entanglement, and enable realizationsand studies of new states of matter, even away from equilibrium. Thesesystems also form the basis for the realization of quantum informationprocessors. While basic building blocks of such processors have beendemonstrated in systems of a few coupled qubits, increasing the numberof coherently coupled qubits to perform tasks that are beyond the reachof modern classical machines is challenging. Furthermore, currentsystems lack coherence and/or quantum nonlinearity for achieving fullyquantum dynamics.

Neutral atoms can serve as building blocks for large-scale quantumsystems. They can be well isolated from the environment, enablinglong-lived quantum memories. Initialization, control, and read-out oftheir internal and motional states is accomplished by resonance methodsdeveloped over the past four decades. Arrays with a large number ofidentical atoms can be rapidly assembled while maintaining single-atomoptical control. These bottom-up approaches are complementary to themethods involving optical lattices loaded with ultracold atoms preparedvia evaporative cooling, and generally result in atom separations ofseveral micrometers. Controllable interactions between the atoms can beintroduced to utilize these arrays for quantum simulation and quantuminformation processing. This can be achieved by coherent coupling tohighly excited Rydberg states, which exhibit strong, long-rangeinteractions. This approach provides a powerful platform for manyapplications, including fast multi-qubit quantum gates, quantumsimulations of Ising-type spin models with up to 250 spins, and thestudy of collective behavior in mesoscopic ensembles. Short coherencetimes and relatively low gate fidelities associated with such Rydbergexcitations are challenging. This imperfect coherence can limit thequality of quantum simulations, and can dim the prospects for neutralatom quantum information processing. The limited coherence becomesapparent even at the level of single isolated atomic qubits.

The present disclosure describes embodiments relating quantum computing.According to some embodiments, methods and systems for quantum computinginvolve first trapping individual atoms and arranging them intoparticular geometric configurations of multiple atoms, for example,using the disclosed acousto-optic deflector system and technique.Systems and methods that allow for precise placement of individual atomsassist in encoding a quantum computing problem. Next, one or more of thearranged atoms may be excited into a Rydberg state, which, as describedbelow, produces particular interactions between the atoms in the array.Next, the system may be evolved. Finally, the state of the atoms may beread out in order to observe the solution to the encoded problem.According to some embodiments, the system and methods described hereinrelate to (1) arranging atoms into 1D and/or 2D arrays (see Section 1),(2) exciting the atoms into Rydberg states and evolving the array ofatoms, for example, using the laser manipulation techniques and highfidelity laser systems described herein (see Section 2), and (3)observing the resulting final state (see Section 3). In addition,additional refinements to the system and methods described herein, suchas to provide a high fidelity and coherent control of the assembledarray of atoms are described in Section 4. Moreover, in Section 5,exemplary problems are discussed which can be solved using the systemand methods described herein.

Section 1: Arranging Atoms into 1D and/or 2D Arrays

According to some embodiments, the states and positions of neutral atomsmay be tightly controlled using focus lasers in a vacuum in order toarrange atoms into arrays for encoding problems solvable by quantumcomputing. A system and method thereof described in the presentdisclosure provides for control of larger number of atoms thanpreviously reported, which vastly expands the set of problems solvableby such quantum systems. These atoms may be prepared in 1D or 2D arraysusing, for example, acoustically excited crystal(s) and laser(s).Systems and methods described in the present disclosure allow for finecontrol of the position of the individual atoms in these 1D or 2Darrays.

The initial states of the atoms can be prepared according to techniquesdescribed in the present disclosure, and the system can then beadiabatically evolved to produce a solution. Accordingly, the system maybe prepared in the lowest-energy state for given initial parameters,which are then evolved to their final values sufficiently slowly for thesystem to remain in the instantaneous lowest-energy state. Furthermore,according to some embodiments, such as described in conjunction withFIGS. 7-10, particular laser control techniques allow for high-fidelityand coherent control of individual atoms after they are arranged into 1Dor 2D arrays. Such techniques allow for greater control over the quantummechanical system, which therefore produces more accurate solutions toencoded problems.

Section 1.A: Acousto-Optic Deflector for Arranging Atoms into 1D Arrays

According to an embodiment, a positioning system and method provides forthe preparation of a large number of atoms, for example, 51 or more, tobe encoded with an initial problem. A laser may pass through a crystal,which may be controlled with one or more discrete acoustic tones (tonefrequencies which vibrate the crystal) to create discrete deflections ofthe laser path corresponding to the applied tone frequencies. The numberof deflections may be controlled by the number of tone frequenciesprovided to the crystal. Tone frequencies are electronic radio-frequencysignals in the VHF band, chosen between, for example, 75 MHz and 125MHz. Tone frequencies can include acoustic waves that are narrow infrequency. Multiple tone frequencies may be overlayed to produce asignal comprising multiple tones. These tone frequencies may then beapplied to the crystal to cause compression or vibration of the crystal.According to some embodiments, these tone frequencies may be provided tothe crystal using, for example, one or more piezoelectric transducersthat may be bonded to the crystal. According to some embodiments, thetone frequencies may be chosen based on the acoustic and/or otherproperties of the crystal. Furthermore, adjusting the frequency of eachindividual tone may adjust the amount of deflection for each discretedeflection, thereby creating a controllable spacing between eachdeflected beam of light. Tone frequencies may be converted from adigital waveform, produced, for example, by a computer controller, to ananalog signal by a software-defined radio (SDR) or an arbitrary waveformgenerator may be implemented that synthesizes the superposition of alldesired tone frequencies in the time domain. The frequencies may beadjusted by updating the digital waveforms output by the computercontroller.

The tone frequencies used depend on a number of factors, such as, butnot limited to, the number of deflections desired, the particularcrystal being used, the wavelength of the light applied to the crystal,and the desired spacing of the confinement regions/traps. The range offrequencies of the acoustic waves applied to the crystal may have limitsbased on the speed of sound in the crystal material, and may be, forexample, between 50-500 MHz. According to some embodiments, a set ofdiscrete frequencies in the range of approximately 75-125 MHZ may beused. According to some embodiments, 100 traps may be generated byspacing individual tone frequencies apart by approximately ˜0.5 MHz.According to some embodiments, when adjusting individual tonefrequencies to the spacing of the traps, the angle as a function ofapplied acoustic frequency may be on the order of approximately 0.1 to10 mrad/MHz. One exemplary value may be approximately 1.2 mrad/MHz.However, this value is merely exemplary, and changes drastically basedon the particular crystal and optics used.

The output laser array beams from the crystal may be focused into acloud of cooled atoms. The atoms may be cooled by the radiation pressureof additional counter-propagating laser beams, with a magneticquadrupole field to induce a spatially dependent radiation pressure thatcounters the atomic motion in all directions and produces a restoringforce to the center of the quadrupole field. The output laser array maybe focused such that each laser array beam may only trap a single atom.The cloud may then be dispersed, leaving only trapped atoms. An imagemay then be taken of the atoms in the trap, e.g., based on lightscattering by the atoms. In a measurement and feedback procedure, thetrapped atoms may then be correlated with individual of the tonefrequencies provided to the crystal, for example, by once averaging amultitude of images of atoms loaded in the traps. The establishedpositions may be recorded and assigned to individual tone frequencies.In subsequent loading of atoms into the traps, tone frequencies withoutassociated trapped atoms may then be turned off after taking an imageand locating trap positions where atomic fluorescence is absent. Each ofthe remaining tone frequencies (e.g., those that have not been turnedoff) may then be swept by adjusting each tone frequency to position theremaining trapped atoms. Thus, according to some embodiments, even ifeach confinement region does not trap an atom, the system may beadjusted so as to reposition the confinement regions to form the desiredarray spacing. Such techniques and systems can significantly improve thenumber of atoms that may be reliably trapped in an array, such as 51atoms or more, and allow for accurate control of the spaced atoms. Afterarranging the atoms in the array, the system may be excited and evolvedas discussed in more detail below, and the resulting changes observed inorder to read out a solution to an encoded problem.

FIGS. 1C-1E show an experimental protocol and setup for arranging andevolving an array of atoms, according to an embodiment. FIG. 1E shows aschematic of a system for arranging and controlling an array of atoms,according to some embodiments. As shown in FIG. 1E, the system mayinclude a crystal 102, a tweezer laser source 106, an acoustic tonegenerator 104, and manipulation laser sources 108A, 108B. Acoustic tonegenerator produces one or more (n) tone frequencies which are applied tothe crystal 102. The tweezer laser source 106 supplies light to thecrystal 102, which is then deflected into n separate tweezer beams thatform a tweezer array 107, each associated with one of the one or moretone frequencies. The frequency of each individual tone frequencydetermines the deflection of the respective tweezer beam. The tweezerbeams may be used to trap atoms 190. The individual tone frequencies maybe adjusted in frequency in order to adjust the spacing of the atoms190. Atoms 190 may then be manipulated by manipulation laser sources108A and 108B in order to evolve the system.

First, as shown in FIG. 1D, in step 110, atoms are loaded from amagneto-optical trap (not shown) into a tweezer array 107 created by anacousto-optic deflector (AOD) including, for example crystal 102 andacoustic tone generator 104. For example, as discussed above, a cloud ofatoms may be prepared, for example in a magneto-optical trap. An AOD mayprovide a plurality of tone frequencies via acoustic tone generator 104to crystal 102 to create a 1D tweezer array 107. Each laser tweezer isassociated with one discrete tone frequency applied to the crystal.Then, the atoms may be brought to the tweezer array 107 (or vice versa)in order to trap individual atoms 190 in the tweezers associated withtone frequencies applied to the crystal 102. As shown in FIG. 1C, thespacing of the loaded atoms may be arbitrary for at least two reasons.First, not every tone frequency is guaranteed to trap an atom from thecloud. Thus, certain tone frequencies may not be associated with atoms.Second, the tone frequencies need not be prepared in a specificallyordered state in order to trap atoms. Thus, the atoms 190 may be spacedat arbitrary frequencies (and thus arbitrary relative spacings) beforeand during the loading process. After trapping the atoms in the AODtweezers, the cloud of atoms may be dispersed. A measurement andfeedback procedure discussed above may be used to eliminate the entropyassociated with the probabilistic trap loading and results in the rapidproduction of defect-free arrays with over 50 laser cooled atoms.

Next, at step 120, the trapped atoms 190 may be prepared in apreprogrammed spatial configuration in a well defined internal groundstate g. As discussed above, each atom in the array 190 may beassociated with one of the tone frequencies applied to the crystal 102,though not all of the resulting traps may be occupied by trapped atoms.The trapped atoms may be imaged, and the locations of the atoms may becorrelated to tone frequencies. Occupied tone frequencies may be notedand maintained, while unoccupied tone frequencies may be cut from theacoustic signal from acoustic tone generator 104 applied to the crystal102. Next, the occupied tone frequencies may be adjusted in order torearrange the atoms 190, for example, into the patterns shown in thearrange row of FIG. 1C. Because the relative spacing of each deflectedlaser beam in the tweezer array 107 (and thus each tweezer) is dependenton the particular tone which causes that deflection, the relativespacing of the tweezers in the tweezer array 107 may be adjusted byadjusting the individual tone frequencies supplied by acoustic tonegenerator 104. Thus, each trapped atom 190 may be repositioned in 1D byadjusting its associated tone frequency. As shown in FIG. 1C, thearbitrarily spaced atoms in the load position can be arranged in aparticular pattern, such as in evenly spaced groups of seven atoms,using this technique.

In step 114, the optical traps or tweezers may then be turned off, forexample by shutting off tweezer laser source 106, to let the systemevolve under the unitary time evolution U (Ω, Δ, t) in a sufficientlyshort time as to neglect atomic motion while avoiding detrimentaleffects of the trapping potential on the unitary evolution U(t). Theevolution U(t) may be realized by coupling the atoms to the Rydbergstate |r

=|70S_(1/2)

with laser light along the array axis, as shown in FIGS. 1A and 1 n step116. As shown in FIG. 1E, laser light from one or more manipulationlasers 108A, 108B may be applied to the atom array 190 during evolution.This is shown in FIG. 1C as the application of the function U(t).Various control techniques are discussed throughout the presentdisclosure, for example with reference to FIGS. 7-10 below. The finalstates of individual atoms are detected in step 118 by turning the trapsback on, and imaging the recaptured ground state atoms via atomicfluorescence using, for example, a camera, while the anti-trappedRydberg atoms are ejected. This is shown in the detect line of FIG. 1C,where dots represent atomic fluorescence associated with present groundstate atoms and circles mark sites where atoms are lost owing to Rydbergexcitation.

Section 1.B: Two-Dimensional Ordering of Atoms for Encoding More ComplexProblems

According to some embodiments, the techniques described in the presentdisclosure can be applied and adapted to prepare arrays of severalhundred individual atoms arranged in a 2D geometry. 2D arrays of atomscan be used to solve a wider range of problems than 1D arrays. Whilesome of the techniques and systems described in the present disclosurereference 1D arrays, they can be applied to 2D arrays using the 2D arraysystems and methods described below. Combined with the laser controlsystem and methods for controlling and manipulating atoms into Rydbergand other quantum states described below, quantum optimizationalgorithms can be implemented to solve real-world problems, such as, butnot limited to the maximum independent set optimization problemsdescribed in the present disclosure.

According to some embodiments, systems and methods described in thepresent disclosure provide for the creation of a large number of trapsin 2D. When atoms are loaded into these traps, they are loaded into eachtrap with finite probability of ˜0.5. A procedure performed in 1D canthen sort the atoms after identifying their locations. According to someadditional embodiments, this sorting (or ‘rearrangement’) procedure canbe applied for 2D arrays of traps.

Crossed-AODs: According to some embodiments, multiple copies of a singlebeam in one direction may be created to generate a 1D pattern using anAcousto-Optic Deflector (AOD) (see above for an explanation of operationof an AOD). Then, a second AOD can be used to create copies of theone-dimensional system in a different (for example, perpendicular)direction to generate a 2D pattern. Other orientations are contemplated.Next, atoms may be loaded into this pattern and defects removed bysuccessively turning off the frequencies in either of the AODs thatcorrespond to rows or columns containing the largest number of emptytraps. Next, the frequencies may be modified on both AODs to reshape thefully loaded pattern into the target 2D array.

As shown in FIG. 14, two AODs 1410 and 1420 may be placed close to oneanother and used to generate a 2D set of traps, according to someembodiments. Each AOD 1410 and 1420 may have acoustic drivers which aredriven by RF signals 1450A and 1450B, respectively, to split theincident light 1440 according to the specific tone frequencies applied.As the 1D light array output from AOD 1410 becomes incident on AOD 1420,AOD 1420 may further split each beam in the 1D array into additionalbeams in another direction. As shown in FIG. 14, the AODs 1410 and 1420may be held at a relative angle to one another, for example, 90 degrees.The relative angle between the two AODs 1410 and 1420 determines therelative orientation of the “rows” and “columns” of the 2D trappingpattern. For instance, 2 AODs 1410 and 1420 perpendicular to one anothercan create a square pattern 1430; 2 AODs with a relative angle of 60degrees can create a triangular pattern (not shown). Such a procedurecan create any geometry in which all “rows” are identical to oneanother, and all “columns” are identical to one another. Atoms may beloaded into the set of traps generated using this technique. Frequenciesassociated to individual “rows” (AOD1) and “columns” (AOD2) may beturned off in order to ensure that the remaining traps contain atoms inthe desired pattern. The set of frequencies in each of the AODs can bemodified to transport the atoms to a final configuration of arbitrarydistances between “columns” and between “rows”.

FIGS. 15A-15D show a procedure for trapping a plurality of atoms in a 2Darray using the system shown in FIG. 14, according to some embodiments.In particular, FIGS. 15A-15D show a procedure to create a 3×3 array oftraps all of which are filled with atoms. It should be appreciated thatthe similar techniques may be used to create other configurations, suchas somewhere some spaces are left empty. FIG. 15A shows the output ofAODs 1410 and 1420 at a right angle to produce a square trapconfiguration 1510 a having 6 rows and 6 columns. The position of thetraps is indicated by the intersection of straight lines 1520 in squarepattern 1510, and atoms are indicated by filled circles 1530. FIG. 15Bshows an array 1510B with rows and columns 1540 marked with “Xs” thatmay be removed while leaving each remaining row and column with threetrapped atoms. FIG. 15C shows array 1510C without rows and columns 1540,which may be dropped as described herein by dropping associated tonefrequencies applied to one of AODs 1410 and 1420. Frequencies of theremaining rows and columns may be adjusted to cause motion in directions1550 to create an even spacing of a 3×3 array of atoms 1510D, as shownin FIG. 15D. It should be appreciated that these techniques may bescaled up to larger arrays and may be used to create configurationswhere not all intersections of lines (i.e., traps) are filled. Accordingto some embodiments, rows and columns are paired so as to control theposition of exactly one atom at the intersection point. Atoms may bearranged in a desired pattern by adjusting the frequencies of theassociated rows/columns.

FIG. 15E shows an instance of a randomly loaded 8×8 array. Thefluorescence of atoms is shown as dark spots, in an underlying patternthat can be inferred from the position of the atoms present. FIGS.15F-15H show a randomly loaded 2×40 array, where the fluorescence oftrapped atoms is shown by the dark regions. The crosses in FIG. 15Gindicate all the “columns” which are turned off. A final fluorescenceimage in FIG. 15H shows the atoms as dark regions in their new positionsafter being rearranged into a 2×18 fully loaded array.

Row-by-row rearrangement in two pairs of crossed AODs: According to someembodiments, a two sets of AOD pairs as described above can be used togenerate a two-dimensional array of tweezers to load a 2D array. Asshown in FIG. 16, two AOD pairs 1600A and 1600 B may be used. AOD pair1600A may be include AODs 1610A, 1620A close to one another is used togenerate a 2D set of traps 1630A. The relative angle between the twoAODs 1610A, 1620A determines the relative orientation of the “rows” and“columns” of the 2D trapping pattern 1630, as discussed above. Atoms maybe loaded into the set of traps generated in this way as discussed abovewith reference to AODs 1410 and 1420. Another pair of AODS 1600B (pair2) is used to generate a different set of traps 1630B using AODs 1610B,1620B. These traps can be overlapped with the traps generated withpair 1. This can be accomplished, for instance, by using an opticalelement 1660 such as, but not limited to, a semi-reflective surface (forexample, a non-polarizing beam splitter), a surface which transmits onelight polarization and reflects the perpendicular one (for example, apolarizing beam splitter), an optical element that transmits light atsome wavelengths and reflects at others (for example, dichroic), or byhaving both sets of traps come from different directions and meet attheir focal points. The set of traps used for rearrangement (pair 2) canbe adjusted to create a stronger confinement than those used to load(pair 1), for instance, by having more optical power per beam, having awavelength closer to the atomic resonance, or having a smaller focus(not extensive list, and combinations of these effects can be used).FIG. 17A shows an array of traps formed via AOD pair 1600A, where trapsare line intersections and dots signify trapped atoms. Overlapping aline of traps (circles in FIG. 17B) generated with AOD pair 1600B on topof a “row” or “column” from the traps generated with AOD pair 1600A canallow all atoms within that “row” or “column” to be primarily controlledby the traps generated with AOD pair 1600B. Additionally, according tosome embodiments, once traps from AOD pair 1600B have been overlappedwith a “row” or “column”, it is possible to turn off that particular“row” or “column” in AOD pair 1600A. The traps generated with AOD pair1600B can now be rearranged using the procedure described for the 1Dcase (see FIG. 17C). After rearrangement, if the particular “row” or“column” from AOD pair 1600A which had been turned off, row 1 of AODpair 1600A can then be turned back on. Turning off the traps generatedwith AOD pair 1600B can now allow the atoms to be held in their newpositions by the underlying traps generated with AOD pair 1600A (seeFIG. 17D). This procedure can now be repeated for a different “row” or“column” of traps generated with AOD pair 1600A, by changing thefrequencies associated to the AODs in AOD pair 1600B, so that a new setof traps can be generated with AOD pair 1600B, which overlap with thenew target “row” or “column” of traps generated with AOD pair 1600B.FIG. 17E shows the array after all rows have been shifted to the leftusing this procedure. It should be appreciated that where manipulationby AOD pair 1600B is only required in 1D, AOD pair 1600B may be replacedby a single AOD. This AOD (or AOD pair) may be referred to as a “controlAOD” while the AOD pair 1600A that is used to maintain the traps may bereferred to as a “hold AOD.” According to some embodiment, the hold AODmay be treated as a set of pixels to be filled by control of the controlAOD so as to create any desired pattern with resolution based on thehold AOD. According to some embodiments, trapped atoms are moved basedon other configurations of the control AOD that do not correspond toentire rows or columns (for example, a square having atoms at eachedge).

A similar procedure is shown with respect to FIGS. 18A-18H, but withmovement of single atoms via a control AOD. As shown in FIG. 18A, a holdAOD has an initial set of atoms trapped therein. As shown in FIG. 18B,an individual atom is selected by the control AOD (shown with a circle).FIG. 18C shows the individual atom is moved by the control AOD, and thenreleased by turning off control AOD in FIG. 18D. In FIG. 18E, anotheratom is selected, moved in FIG. 18F, and then released in FIG. 18G. FIG.18H shows the array trapped in hold AOD after arrangement. The resultingarray of atoms may have a higher yield.

According to some embodiments, the methods described above could also beconducted without turning off rows or columns of the hold AOD. Instead,the control AOD may be formed with deeper (stronger) traps than the holdAOD. Thus, when a trap of the control AOD that is overlapping a hold AODtrap is moved, the atom will follow the deeper trap of the control AODas if it is not trapped by the hold AOD. However, if moved to anotherhold AOD trap followed by a shutdown of the control AOD trap, the atommay remain trapped by the hold AOD.

Stacked AODs: According to some embodiments, multiple (N) AODs may bestacked on top of one another. Multiple AODs can be used in parallel togenerate independent 1D sets of traps in which to trap and rearrangeatoms. FIGS. 19A and 19B show an exemplary system to combine thedifferent 1D sets of traps into a 2D pattern. As shown in FIG. 19A, aset of 5 AODs 1910 viewed from a top angle are aligned so as to sendlight arrays to multifaceted reflective surface 1920. The light is thenreflected of the multifaceted reflective surface 1920 to lens 1930,which focuses the beams of light. FIG. 19B shows the same elements witha side view. Multifaceted reflective surface 1920, can be, for example,a polished reflective substrate, a combination of several reflectivesurfaces, the previous two but with a non-reflective substrate coated ina reflective material, a deformable mirror or mirror array (list is notextensive. Each 1D set of traps from AODs 1910 can be redirected viamultifaceted reflective surface 1920. A transmissive multifacetedsurface with controllable index of refraction may also be used, such as,a dielectric of variable density or width, or a patterned dielectricwith holes or with alternating sections with distinct indices ofrefraction, or other suitable surfaces. Such a structure can be used tocreate a wavefront equivalent to that of multiple beams overlapping in awell-defined volume. Lens 1930 may be used to focus all beams onto adesired plane. Using N independent AODs 1910, each creating anindependent 1D pattern of traps, it is possible to load atoms into suchtraps, and rearrange each independent 1D set of traps in a desired wayto position atoms in a desired pattern. The rearrangement of all 1D setsof traps can happen simultaneously, or in any desired order.

FIGS. 20A-20C show methods of operating the system of FIGS. 19A-19B,according to some embodiments. As shown in FIG. 19A, a 6×6 array oftraps is produced. N rows correspond to 1D AODs 2010A-2010N (in thiscase 6). The independent 1D sets of traps from AODs 2010A-2010N areindicated by small empty circles connected by a straight line and atomsare indicated by filled circles. As shown in FIG. 19B, the square 6×6array of traps is randomly loaded. As shown in FIG. 19C, the traps arerearranged to collect all atoms on the left of each 1D set of traps.Rearrangement may occur by adjusting the tone frequencies associatedwith the traps for each of AODs 2010A-2010N. It should be appreciatedthat these individual AOD arrays may be implemented as the controland/or hold AODs as described above.

Trap generation in SLM and rearrangement with crossed AODs: According tosome embodiments, a spatial light modulator (SLM) may be used togenerate a 2D array of traps. The SLM can be used to modify thewavefront of a light beam to generate arbitrary spatial patterns oflight intensity that may be used in place of a hold AOD as describedabove as a hold trap array. There exist different types of SLMs, whichaffect the local intensity (example: digital micromirror device, DMD),phase (Liquid Crystal On Silicon, LCOS), or both, of the transmitted orreflected light field to alter its wavefront in a programmable way.Using such a device, a 2D pattern of traps with arbitrary geometries canbe generated, and atoms can be loaded into the set of traps generated inthis way, such as via a control AOD or control AODs as discussed above.FIG. 21A shows an embodiment where a reflective SLM 2110A is used as ahold SLM, and a pair of AODs 2120A is used to manipulate the position ofatoms in the array 2130A. FIG. 22B shows an embodiment where atransmissive SLM 2110B is used as a hold SLM, and a pair of AODs 2120Bis used to manipulate the position of atoms in the array 2130B.According to some embodiments, the pair of AODs 2120A or 2120B can beused to generate a different set of traps that are loaded with atoms.The traps 2130A or 2130B shown in FIGS. 22A and 22B can be overlappedwith the traps generated with the SLM. This can be accomplished, forinstance, by an element 2140, such as a semi-reflective surface(example: non-polarizing beam splitter), a surface which transmits onelight polarization and reflects the perpendicular one (example:polarizing beam splitter), an optical element that transmits light atsome wavelengths and reflects at others (example: dichroic), or byhaving both sets of traps come from different directions and meet attheir focal points. The set of traps used for rearrangement generatedwith the AOD 2120A or 2120B can be adjusted to create a strongerconfinement than those generated with the SLM 2110A or 2110B used toload, for instance, by having more optical power per beam, having awavelength closer to the atomic resonance, and/or having a smaller focus(not extensive list, and combinations of these effects can be used).Overlapping a line of traps generated with the AOD pair 2120A or 2120Bon top of a subset of the traps generated with SLM 2110A or 2110B, canallow all atoms within that subset to be primarily controlled by thetraps generated with the AOD pair. The traps generated with the AOD pair2120A or 2120B can thus be rearranged within the 2D plane of traps tooverlap them either simultaneously, or sequentially, with other trapsgenerated by the SLM 2110A or 2110B. Turning off the specific trapsgenerated with the AOD pair 2120A or 2120B, while they are overlappedwith other traps generated by the SLM 2110A or 2110B, can allow theatoms to be held in their new positions by the underlying trapsgenerated the SLM 2110A or 2110B. This procedure, which mimics thatshown in FIGS. 17A-18D or 18A-18H if the SLM 2110A or 2110B is regularin spacing between traps, can be repeated for a different subset oftraps generated by the SLM 2110A or 2110B, by changing the frequenciesassociated to the AOD pair, so that a new set of traps can be generatedwith the AOD pair 2120A or 2120B, which overlap with the new targetsubset of traps generated with the SLM 2110A or 2110B.

According to some embodiments, the position of the traps in the SLM2110A or 2110B includes separate arbitrary and regular positions. FIG.22A shows a pattern generated by the arbitrary portion of SLM 2110A or2110B, which is indicated by small empty circles near the top.Furthermore, the array of atoms trapped in the regular portion of SLM2110A or 2110B is shown below (the distance between the two may or maynot be to scale). The position of the traps in the regular array isindicated by the intersection of straight lines and atoms are indicatedby filled circles. According to some embodiments, a control AOD may beused (shown as large empty circles) to reposition the atoms in thearbitrary array as shown in FIGS. 22A-22D, and to take atoms from theregular array and move them to the arbitrary array for repositioning asshown in FIGS. 22E-22P. Note that multiple atoms may be moved from theregular portion for placement in the irregular portion and the sametime, as shown in FIGS. 22E-22F and 22L-22M. After moving to theirregular portion, the horizontal spacing between the controlled atomsmay first be adjusted, as shown in FIGS. 22F and 22M-N. FIG. 26 shows anexemplary instance of a 30×50 regular array of traps generated with areflective LCOS-SLM, with the corresponding phase pattern imprinted onthe wavefront of reflected light beam by the LCOS-SLM. FIG. 27 shows acorresponding spatial phase pattern added to the laser light field totransform a single input beam into a 30×50 array of traps

Generate optical lattice and use traps to rearrange atoms within it:According to some embodiments, a 2D trap array may instead be generatedby using a large-lattice-spacing optical lattice. The interference fromtwo light sources can create patterns of light intensity, which can beused to trap cold neutral atoms, which are called optical lattices.These traps may function in place of the hold AOD as a hold trap array.Thus, the optical lattice can be used to hold trapped atoms and combinedwith, for example, a control AOD as discussed above in order torearrange atoms within the optical lattice, such as with the methods andsystems described with respect to FIGS. 17A-17E (shown similarly inFIGS. 24A-24E) and 18A-18H (shown similarly in FIGS. 25A-25H). As shownin FIG. 23, light sources 2320 may create an interference form 2330 thatforms a hold trap array. AODs 2310A, 2310B may be used to manipulatetrapped atoms in the interference form 2330. The pair of AODS is used togenerate a set of traps. These traps can be overlapped with the trappingregions of the optical lattice. This can be accomplished, for instance,by using a semi-reflective surface (example: non-polarizing beamsplitter), a surface which transmits one light polarization and reflectsthe perpendicular one (example: polarizing beam splitter), an opticalelement that transmits light at some wavelengths and reflects at others(example: dichroic), or by having both sets of traps come from differentdirections and meet at their focal points. The set of traps used forrearrangement can be adjusted to create a stronger confinement than thatprovided by the optical lattice used to load, for instance, by havingmore optical power per beam, having a wavelength closer to the atomicresonance, or having a smaller focus (not extensive list, andcombinations of these effects can be used). Overlapping a line of trapsgenerated with the AOD pair on top of a “row” or “column” from theoptical lattice, can allow all atoms within that “row” or “column” to beprimarily controlled by the traps generated with the AOD pair. The trapsgenerated with the AOD pair can be rearranged using the proceduredescribed for the 1D case. Turning off the traps generated with the AODpair can allow the atoms to be held in their new positions by theunderlying optical lattice. This procedure can be repeated for adifferent “row” or “column” of the optical lattice, by changing thefrequencies associated to the AODs, so that a new set of traps can begenerated with the AOD pair, which overlap with the new target “row” or“column” of the optical lattice.

Section 2. Excitation and Evolution

According to an embodiment, the arranged atom arrays may then be excitedand evolved to compute the answer to the encoded problem. Lasers withphoton energy approximately equal to a transition energy of an outermostelectron of the atoms may be used to excite the outermost electrons inthe atoms so as to transition the atoms into an excited state.Particular laser control and application techniques are described inmore detail in the present disclosure. Interactions between the atomsmay be so strong that only some of the atoms, and in particular, onlysome of the atoms in particular regions may transition into an excitedstate. For example, proximity to another excited atom may increase theexcitation energy of a nearby non-excited atom such that a transition ofthe nearby atom is unlikely. The likelihood of a transition of the atomsmay be controlled initially by the distances between individual atoms.According to an embodiment, the exited atoms may be diffused away fromthe traps, and the remaining atoms may be imaged in order to determinewhich of the atoms did not become excited. This final result can producea solution to the encoded problem.

According to an embodiment, atom-by-atom assembly may be used todeterministically prepare arrays of individually trapped cold neutral⁸⁷Rb atoms in optical tweezers. As show in FIG. 1A, controlled, coherentinteractions between atoms 190 may be introduced by coupling them toRydberg states. This results in repulsive van der Waals interactions(Vi=C/R⁶, C>0) between Rydberg atom pairs at a distance Rij. The quantumdynamics of this system is described by the following HamiltonianEquation (1):

$\begin{matrix}{\frac{\mathcal{H}}{\hslash} = {{\sum\limits_{i}{\frac{\Omega_{i}}{2}\sigma_{x}^{i}}} - {\sum\limits_{i}{\Delta_{i}n_{i}}} + {\sum\limits_{i < j}{V_{ij}n_{i}n_{j}}}}} & (1)\end{matrix}$

where Ωi are the Rabi frequencies associated with individual atoms, Aiare the detunings of the driving lasers from the Rydberg state (see FIG.1B), σi=|gi

ri|+|ri

gi| describes the coupling between the ground state |g

and the Rydberg state |r

of an atom at position i, and ni=|ri

ri|. In general, within this platform, control parameters Ωi, Δi may beprogrammed by changing laser intensities and detunings in time.According to an embodiment, homogeneous coherent coupling may be used(|Ωi|=Ω, Δi=Δ). The interaction strength Vij may be tuned by eithervarying the distance between the atoms or carefully choosing the desiredRydberg state.

The ground state |g

and the Rydberg state |r

can be used as qubit states to encode quantum information. The coherentcoupling between these states is provided by the laser light and allowsfor manipulation of the qubits. Furthermore, the Rydberg states ofmultiple atoms strongly interact with each other, enabling engineered,coherent interactions. These strong, coherent interactions betweenRydberg atoms can provide an effective constraint that preventssimultaneous excitation of nearby atoms into Rydberg states. FIG. 1Fshows such an effect, which is also sometimes called Rydberg blockade.When two atoms are sufficiently close so that their Rydberg-Rydberginteractions Vij exceed the effective Rabi frequency Ω, then multipleRydberg excitations can be suppressed. This provides the Rydbergblockade radius, Rb, for which Vij=Ω (Rb=9 μm for |r

=|70S

and Ω=2π×2 MHz as used here). In the case of resonant driving of atomsseparated by a distance of a=24 μm, we observe Rabi oscillationsassociated with non-interacting atoms as shown in the top curve of FIG.1F. However, the dynamics change significantly as we bring multipleatoms close to each other (a=2.95 μm<Rb). In this case, Rabioscillations between the ground state and a collective W-state withexactly one excitation ˜_(i)Ω_(i)|g₁ . . . r_(i) . . . g_(N)

with the characteristic N^(1/2)-scaling of the collective Rabi frequencycan be observed. These observations allow quantification of thecoherence properties of the system. In particular, the contrast of Rabioscillations in FIG. 1F is mostly limited by the state detectionfidelity (93% for r and 98% for g). The individual Rabi frequencies anddetunings are controlled to better than 3% across the array, while thecoherence time is ultimately limited by the probability of spontaneousemission from the state |e

during the laser pulse (scattering rate 0.022/μs).

As shown in FIG. 1A, individual ⁸⁷Rb atoms are trapped using opticaltweezers and arranged into defect-free arrays. Coherent interactions Vijbetween the atoms are enabled by exciting them to a Rydberg state, withstrength Ω and detuning Δ.

FIG. 1B shows a two photon process can be used to couple the groundstate |g

=|5S_(1/2), F=2, mF=−2

to the Rydberg state |r

=|71S_(1/2), J=½, m_(J)=−½

via an intermediate state |e

=|6P_(3/2), F=3, m_(F)=−3

using circularly polarized 420 nm and 1013 nm lasers (typically δ˜2π×560MHz >>Ω_(B), Ω_(R)˜2π×60, 36 MHz).

As shown in FIG. 1C, the experimental protocol may include loading theatoms into a tweezer array (1) and rearranging them into a preprogrammedconfiguration (2). After this, the system may evolve under U (t) withtunable parameters Δ(t), Ω(t), Vij. This can be implemented in parallelon several non-interacting sub-systems (3). The final state can bedetected by a suitable technique, such as using fluorescence imaging(4).

As shown in FIG. 1F, for resonant driving (Δ=0), isolated atoms (toppoints) display Rabi oscillations between |g

and |r

. Arranging the atoms into fully blockaded clusters N=2 (as shown in themiddle plot) and N=3 (as shown in the bottom plot) atoms result in onlyone excitation being shared between the atoms in the cluster, while theRabi frequency is enhanced by N^(1/2). Multiple excitations (middle andbottom points) are strongly suppressed. Error bars indicate 68%confidence intervals (CI) and are smaller than the marker size.

Section 2.A: Exemplary Control and Evolution of 1D Arrays of Atoms

Finding ways to engineer and control large quantum systems is a majorchallenge to quantum computing. The control and evolution examplesdiscussed below allow for arrays of up to 51 atoms or more, which canserve as qubits, with a very large amount of controllability andengineered, coherent interactions between them. Furthermore, asdiscussed in the present disclosure, this system lends itself well forscaling up to larger numbers of atoms as well as controllability at thesingle atom level. The techniques and experiments described in thepresent disclosure show that engineering and controlling such largequantum systems is possible. Such control is required for performingquantum simulations. Such quantum simulations can be used to solve otherreal-world problems, for example finding new materials (a famous exampleis high temperature super conductivity), understanding complex moleculestructures and designing new once. Other applications includeoptimization problems, such as the maximum independent set problemdiscussed in more detail below. These optimization problems map directlyto real world problems.

According to some embodiments, Rydberg crystals, or controlled arrays ofRydberg atoms, can be created as discussed in more detail below. TheseRydberg crystals provide a good test-bed for the large quantum systemsproduced using the methods and systems discussed herein. In general, itis very hard to characterize a large quantum system due to theexponentially increasing complexity as the system size is increased.Since the solution to the ordered state of the Rydberg crystal is known,creation and characterization of Rydberg crystals provide forbenchmarking of the systems and techniques used to create and manipulatethe Rydberg crystals. As discussed below, these systems and methodsdemonstrate coherent control and that this large quantum system (theRydberg crystal) shows a high degree of coherence. In addition, it isdemonstrated that the Rydberg crystals created and controlled using thesystems and methods described herein have special quantum states thatshow surprisingly robust dynamics as they are driven out of equilibrium.This unique property is discussed in more detail below.

According to some embodiments, arranged 1D arrays of atoms may beexcited and evolved to produce solutions to quantum computing problemsand may be used as a quantum simulator. Described below are techniquesfor exciting and controlling a 1D array of atoms, as well ascharacterization of the interaction between the atoms. In the case ofhomogeneous coherent coupling, the Hamiltonian Equation (1) closelyresembles the paradigmatic Ising model for effective spin-½ particleswith variable interaction range. Its ground state exhibits a richvariety of many-body phases that break distinct spatial symmetries, asshown in FIG. 2A. For example, at large, negative values of Δ/Ω itsground state corresponds to all atoms in the state g, corresponding toparamagnetic or disordered phase. As Δ/Ω is increased towards largepositive values, the number of atoms in r rises and interactions betweenthem become significant. This gives rise to spatially ordered phaseswhere Rydberg atoms are regularly arranged across the array, resultingin ‘Rydberg crystals’ with different spatial symmetries as shown in FIG.2A. The origin of these correlated states can be understood by firstconsidering the situation when V_(i,i+1)»Δ»Ω»V_(i,i+2), i.e. blockadefor neighboring atoms but negligible interaction between second-nextneighbors. In this case, the Rydberg blockade reduces nearest-neighborexcitation, while long-range interactions are negligible, resulting in aRydberg crystal breaking Z₂ translational symmetry that is analogous toantiferromagnetic order in magnetic systems. Moreover, by tuning theparameters such that V_(i,i+1), V_(i,i+2)»Δ»Ω»V_(i,i+3) and V_(i,i+1),V_(i,i+2), V_(i,i+3)»Δ»Ω» V_(i,i+4), arrays with broken Z3 and Z4symmetries may be obtained, respectively, as shown in FIG. 2A. The boxedareas 220 in FIG. 2A indicate potential incommensurate phases.

To prepare the system in these phases, the detuning Δ(t) of the drivinglasers may be dynamically controlled to adiabatically transform theground state of the Hamiltonian from a product state of all atoms in ginto crystalline Rydberg states. First, all atoms may be prepared instate |g=5_(S1/2), F=2, m_(F)=2

by optical pumping. The laser fields may then be switched on and sweptthe two-photon detuning from negative to positive values using afunctional form shown in FIG. 3A.

As shown in FIG. 2B, the resulting single atom trajectories in a groupof 13 atoms for three different interaction strengths as the detuning Δis varied. In each of these instances, a clear transition from theinitial state |g₁, . . . , g₁₃

to a Rydberg crystal of different symmetry can be observed. The distancebetween the atoms determines the interaction strength which leads todifferent crystalline orders for a given final detuning. For instance,to achieve a Z₂ order, we arrange the atoms with a spacing of 5.9 μm,which results in a nearest neighbor interaction of V_(i,i+1)=2π×24MHz»Ω=2π×2 MHz while the next-nearest interaction is small (2π×0.38MHz). This results in a buildup of antiferromagnetic crystal where everyother trap site is occupied by a Rydberg atom (Z₂ order). By reducingthe spacing between the atoms to 3.67 μm and 2.95 μm, Z₃₋ and Z₄₋ ordersare respectively observed, as shown in FIG. 2B.

More specifically, FIG. 2B shows the buildup of Rydberg crystals on a13-atom array is observed by slowly changing the laser parameters asindicated by the red arrows in a (see also FIG. 3A). The bottom panelshows a configuration where the atoms are a=5.9 μm apart which resultsin a nearest neighbor interaction of V_(i,i+1)=2π×24 MHz and leads to aZ₂ order where every other atom is excited to the Rydberg state |r

. The right bar plot displays the final, position dependent Rydbergprobability (error bars denote 68% CI). The configuration in the middlepanel (a=3.67 μm, V_(i,i+1)=2π×414.3 MHz) results in Z₃ order and thetop panel (a=2.95 μm, V_(i,i+1)=2π×1536 MHz) in a Z₄ ordered phase. Foreach configuration, a single-shot fluorescence image before (left) andafter (right) the pulse is shown. Open circles highlight lost atoms,which are attributed to Rydberg excitations.

Performance of the quantum simulator may be compared to the measured Z²order buildup with theoretical predictions for a N=7 atom system,obtained via exact numerical simulations. FIGS. 3A-3C shows comparisonwith a fully coherent simulation. As shown in FIGS. 3A-3C, this fullycoherent simulation without free parameters yields excellent agreementwith the observed data when the finite detection fidelity is accountedfor. As shown in FIG. 3A, the laser driving consists of a square shapedpulse Ω(t) with a detuning Δ(t) that is chirped from negative topositive values. FIG. 3B shows time evolution of Rydberg excitationprobability for each atom in a N=7 atom cluster (colored points), whichmay be obtained by varying the duration of laser excitation pulse Ω(t).The corresponding curves are theoretical single atom trajectoriesobtained by an exact simulation of quantum dynamics with (1), with thefunctional form of Δ(t) and Ω(t) used as discussed with respect to thisexample, and finite detection fidelity. FIG. 3C shows evolution of theseven most probable many-body states. The evolution of the many-bodystates in FIG. 3C shows that the perfect antiferromagnetic target statemay be measured with 54(4)% probability. When correcting for the knowndetection infidelity, the desired many-body state is reached with aprobability of p=77(6)%. Error bars denote 68% CI.

Preparation fidelity depends on system size, as shown in FIG. 4 byadiabatic sweeps on arrays of various sizes. FIG. 4A shows Preparationfidelity of the crystalline ground state as a function of cluster size.The open dots are the measured values and the filled dots are correctedfor finite detection fidelity. Error bars denote 68% CI. FIG. 4B shows ahistogram of the number of microstates without correction for detectionerrors per observed number of occurrence in a 51-atom cluster for 18439experimental realizations. The most occurring microstate is the groundstate of the many-body Hamiltonian. The probability of finding thesystem in the many-body ground state at the end of the sweep decreasesas the system size is increased. However, even at system sizes as largeas 51 atoms or more, the perfectly ordered crystalline many-body stateis obtained with p=0.11(2)% (p=0.9(2)% when corrected for detectionfidelity), which is remarkable in view of the exponentially large251-dimensional Hilbert space of the system. Furthermore, as shown inFIG. 4B, this state with perfect Z₂ order is by far the most commonlyprepared state.

Section 3: Observing the Resulting Final State

After a quantum computation the state of the atoms can be detected byfluorescence imaging. This may be done by state dependent atom lossesand subsequent imaging to reveal the remaining atoms. In the examplesdescribed herein, the tweezer potentials may be restored after the laserpulse. Atoms that are in the ground state are recaptured by thetweezers, whereas those left in the Rydberg state are pushed away by thetweezer beams. A subsequent fluorescence detection may reveal the stateof each atom. An extension of this detection may be to first map theRydberg state to a second hyperfine state, after which state selectivefluorescence may be employed to image groups of atoms in each state.This provides the additional advantage that atoms are not lost at theend of the computation.

Section 4. Laser Control System for Manipulating an Array of Atoms

Individual neutral atoms excited to Rydberg states are a promisingplatform for quantum simulation and quantum information processing.However, experimental progress to date has been limited by shortcoherence times and relatively low gate fidelities associated with suchRydberg excitations. Thus, even where the methods described above may beused to assemble a large array of atoms for quantum computing, it isstill necessary to develop high-fidelity and coherent control of theassembled array of atoms in order to evolve arranged arrays of atoms tosolve particular problems. Thus, according to an embodiment,high-fidelity (low-error, such as close to 0% error) quantum control ofRydberg atom qubits can be implemented using the system and methodsdescribed, for example, with respect to FIGS. 7A-7D and/or methodsdescribed above. Enabled by a reduction in laser phase noise, thisapproach yields a significant improvement in coherence properties ofindividual qubits. This high-fidelity control extends to themulti-particle case by preparing a two-atom entangled state with afidelity exceeding 0.97(3) (i.e., an error rate of only 3/100), andextending its lifetime with a two-atom dynamical decoupling protocol.These advances provide for scalable quantum simulation and quantumcomputation with neutral atoms that can more accurately and consistentlymanipulate atoms in an ordered array.

According to some embodiments, high-fidelity quantum control of Rydbergatom qubits can be achieved by reducing laser phase noise, thusproducing a significant improvement in the coherence properties ofindividual qubits. This high-fidelity control extends to themulti-particle case is confirmed by experimental results preparing atwo-atom entangled state with a fidelity exceeding 0.97(3). The lifetimeof the prepared Bell state can also be extended with a novel two-atomdynamical decoupling protocol, according to some embodiments.

FIGS. 7A-7D show and characterize a control system for single-atom Rabioscillations of individual cold Rubidium-87 atoms in optical tweezers atprogrammable positions in one dimension, according to an embodiment. Asshown in FIG. 7A, a laser, such as an external-cavity diode laser (ECDL)710A can provide light to a reference optical cavity (REF) 720. Thelight from ECDL 710A can be locked to the REF 720 using, for example, aPound-Drever-Hall technique. For this, the light of the ECDL may bephase modulated and the reflection of the cavity 720 detected on a photodiode PD 770. This signal is demodulated to create an error signal thatis used to lock the laser. This type of lock can create a very narrowlinewidth. However, it may also produce high frequency phase noise atthe bandwidth of the lock. To mitigate this noise, the cavity Ref 720can be used as a filter cavity, whose narrow transmission window (shadedregion in inset) suppresses high-frequency phase noise To enhance theoutput power of the REF 720, the transmitted light can be used toinjection lock a Fabry-Perot (FP) laser diode 740. This can be achievedby proding the light from REF 720 through an optical isolator (ISO) 730to FP 740. Using this technique, the light from FP 740 will inherit thespectral properties of the light from REF 720 albeit at a higher power.The light from FP 740 can be provided through ISO 730 and focusingoptics 750A to an array of atoms 790. A second laser source 710B canprovide laser light at a different frequency, which can be focusedthrough optics 750B onto the array of atoms 790. In certain embodiments,the light from optics 750A and 750B can be provided in acounter-propagating configuration to minimize the Dopler sensitivity oftransitions. According to some embodiments, it is desirable to ensuregood alignment of the excitation beams relative to the array of atoms790. To achieve this, an active feedback scheme may be employed thatcontrols the alignment of the incoming beams. A small amount of lightfrom optics 750A can be picked off and provided to a spatially resolvedimaging device, such as CCD 760, for alignment. This narrow and accuratecontrol system for the lasers 710A and 710B allows for control of atomsinternal states |g> and |r>. Additional components may be added, such ascooling lasers and magnetic field generation structures such as fieldcoils.

According to some embodiments, the atoms 790 are initialized in a Zeemansublevel |g

=|S_(1/2), F=2, m_(F)=−2

of the ground state via optical pumping in a 1.5 G magnetic field. Thetweezer potential is then rapidly switched off, and a laser field fromboth lasers 710A and 710B is applied to couple the atoms 790 to theRydberg state |r

=|70S, J=½, m_(j)=−½

. After the laser pulse, for example, of duration 3-8 μs, the tweezerpotentials are restored. Atoms 790 that are in the ground state arerecaptured by the tweezers, whereas those left in the Rydberg state arepushed away by the tweezer beams. According to some experimentalembodiments, this detection method has Rydberg state detection fidelityf_(r)=0.96(1) and ground state detection fidelity f_(g) ranging from0.955(5) to 0.990(2), depending on the trap-off time.

According to some embodiments, the Rydberg states are excited via atwo-photon transition. The ECDL 710A may be a 420 nm laser that isdetuned by A above the frequency of the transition from |g

to |e

=|6P_(3/2), F=3, m_(F)=−3. The second laser source 710B provides a laserfield, for example, at 1013 nm couples |e

to |r

. The two lasers 710A and 710B are polarized to drive σ⁻ and σ⁺transitions, respectively, such that only a single intermediate subleveland Rydberg state can be coupled, avoiding the population of additionallevels and associated dephasing. These transitions are shown in FIG. 7B.

The two lasers 710A, 710B may any known lasers, such as external-cavitydiode lasers sold by MOG Laboratories Pty Ltd. The lasers 710A may befrequency stabilized by a Pound-Drever-Hall (PDH) 770, such as a PDHprovided by Stable Laser Systems, lock to an ultra-low expansionreference cavity. Laser 710B may also be stabilized by a similar PDH(not shown). The PDH 770 lock strongly suppresses laser noise atfrequencies below the effective bandwidth of the lock, resulting innarrow linewidths of <1 kHz, as estimated from in-loop noise. However,noise above the lock bandwidth cannot be suppressed, and can beamplified at high locking gain. This results in broad peaks in phasenoise around ˜2π×1 MHz (see inset of FIG. 7A). This high-frequency phasenoise presents a coherence limitation in Rydberg experiments andexperiments with trapped ions. To suppress this phase noise, a referencecavity 730 may be used as a spectral filter. In particular, thetransmission function of the cavity may be a Lorentzian with afull-width at half maximum of Γ˜2π×500 kHz (corresponding to a finesseof F˜3000). Other functions could be used for cavities with differentfunctions. The smaller the linewidth, the better the filtering. When thelaser is locked, its narrow linewidth carrier component is transmittedthrough the cavity, whereas the high-frequency noise at 2π×1 MHz issuppressed by a factor of >4. While only shown in FIG. 7A with relationto ECDL 710A, to amplify this light at both 420 and 1013 nm, the twocolors may be split, and each beam used to injection lock a separatelaser diode, which inherits the same spectral properties. This amplifiesthe spectrally pure transmitted light to 5 mW of 420 nm and 50 mW of1013 nm light. While the 420 nm power is sufficient to drive the bluetransition directly, in some embodiments the 1013 nm may be furtheramplified by a tapered amplifier (not shown).

The lasers 710A, 710B may be provided onto the atom array 790 in acounter propagating configuration to minimize Doppler shifts due tofinite atomic temperature. The lasers 710A, 710B may be focused to awaist of 20 or 30 μm, respectively. According to an experimentalembodiment, single photon Rabi frequencies of Ω_(B)≃2π×60 MHz(Ω_(R)≃2π×40 MHz) can be achieved. At intermediate detuning of Δ≃2π×600MHz, this leads to a two-photon Rabi frequency of Ω=Ω_(B)Ω_(B)/(2Δ)≃2π×2MHz. Each beam is power-stabilized to <1% by an acousto-optic modulatorthat is also used for fast (˜20 ns) switching. To minimize sensitivityto pointing fluctuations, well-centered alignment onto the atoms can beensured using the reference camera 760 and an automatic beam alignmentprocedure, where the beam position is stabilized to a fixed position onthe reference camera using one steering mirror mount with piezoactuators. The optimal position may be established by measuring the Rabifrequency on the atoms for different beam positions on the camera andchoosing the position that maximizes the coupling to the Rydberg state.

FIG. 7D shows a similar system to 7A with the addition of furthercontrol of the laser 710B, according to some embodiments. FIG. 7D showstwo lasers at 420 nm 710A and 1013 nm 710B. Each laser 710A, 710B maybe, for example, an external cavity diode laser. Lasers 710A, 710B areused to coherently excite atoms 790 to Rydberg states. Both lasers arestabilized to a reference optical cavity 720 by Pound-Drever-Hall (PDH)locks. The mirror coatings of the REF 720 may be chosen such that thecavity is suited for multiple wavelengths and may be used by both 710Aand 710B. For this purpose, the lasers are each phase modulated with anelectro-optic modulator (EOM) placed between each laser source and thereference cavity. (not shown), and the light reflected from the opticalcavity is measured on photodetectors (PD) 770A, 770B, respectively, andused to feedback on the laser by tuning the current through the laserdiode. This stabilizes each laser to narrow linewidths of <1 kHz, asmeasured from noise on the error signal of the PDH lock. However, highfrequency noise from the laser diode cannot be suppressed and caninstead be amplified by the PDH lock due to its finite bandwidth. Thisleads to broad peaks in noise at around +/−1 MHz relative to the centralnarrow carrier linewidth (shown in the power spectrum insets).

While the lasers 710A, 710B are locked to the reference cavity, thelight is primarily transmitted through the cavity 720. However, thecavity 720 acts as a low pass filter with a bandwidth of ˜500 kHz, andits transmission therefore suppresses noise outside this ‘transmissionwindow’ (schematically shown as the boxed region with dashed line in thepower spectrum insets). The transmitted light through the cavitytherefore has a narrow linewidth but also suppressed high frequencynoise.

Since high power is beneficial for control of the atoms 790, thetransmitted light of each laser 710A, 710B is split and injection lockedthrough optical isolators (ISOs) 730A, 730B, respectively intoindependent Fabry-Perot (FP) laser diodes 740A, 740B, respectively.These laser diodes 740A, 740B inherit the same high-quality spectralproperties of the light used to seed them, and effectively amplify thisseed light to ˜5 mW of 420 nm light and ˜50 mW of 1013 nm light. The1013 nm power may be additionally amplified by a tapered amplifierpositioned after the laser diode 740B (not shown). The two lasers 710A,710B may then be focused by lenses 750A, 750B onto the array of atoms790 in a counter-propagating configuration to minimize the Dopplersensitivity of the transition.

This laser stabilization and filtering scheme enables high fidelitycoherent control of Rydberg atom qubits to date. The scheme could befurther improved by using a commercially available higher finesseoptical cavity that has a narrower linewidth arising from mirrors ofhigher reflectivity, and therefore a higher factor of noise suppression.According to some embodiments, intrinsically lower noise laser sources,such as Titanium-Sapphire lasers or dye lasers, can be used to drivethis transition without needing to spectrally filter the high frequencylaser phase noise.

Section 4.A: Experimental Results from Improved Laser Control

According to some embodiments, various control methods and systemsdisclosed herein may be implemented to extend coherence times andimprove control over atoms. According to an experimental embodimentimplementing the system and methods described above, such as the controlsystem shown in FIGS. 7A-7D, long-lived Rabi oscillations were measuredto have a 1/e lifetime of τ=27(4) μs, to be compared with a typical

7 μs lifetime in previous experiments. This is shown in FIG. 7C, whichshows resonant two-photon coupling induces Rabi oscillations between |g

=and |r

. Each plot shows the Rydberg probability as a function of time inmicroseconds. The upper plot is a measurement from a setup used in theart. The lower plot shows typical results with the setup describedabove, with a fitted coherence time of 27(4) μs. Each data point iscalculated from 50-100 repeated measurements, averaged over twoidentically-coupled atoms separated by 23 μm such that they arenon-interacting. Error bars are 68% confidence intervals. The solidlines are fits to experimental data, while the dotted lines indicate theexpected contrast from a numerical model. As shown in FIG. 7D, there isexcellent agreement between these new measurements and a simplenumerical model for the single-atom system, indicated by dotted lines inFIG. 7D. The numerical model implemented had no free parameters andaccounts only for the effects of random Doppler shifts at finite atomictemperature, off-resonant scattering from the intermediate state |e

, and the finite lifetime of the Rydberg state |r

. The results from the numerical model are additionally scaled toaccount for detection fidelity.

According to another experimental embodiment, the coherence of singleatoms and single-qubit control can be characterized. To begin, thelifetime of the Rydberg state is measured, as shown in FIG. 8A, in orderto demonstrate and determine the timescale during which quantum controlmay be performed. As shown in FIG. 8A, the lifetime of |r

can be characterized by exciting from |g

to |r

with a π-pulse, and then de-exciting after a variable delay. Theprobability to end in |g

(denoted P_(g)) decays with an extracted lifetime of T₁=51(6) μs. Themeasured T₁=T_(r→g)=51(6) μs is consistent with the 146 μs Rydberg statelifetime when combined with the ˜80 μs timescale for off-resonantscattering of the 1013 nm laser from |e

. A Ramsey experiment shows Gaussian decay 810 that is well-explained bythermal Doppler shifts (see FIG. 8B). As shown in FIG. 8B, a Ramseyexperiment plotted as line 810 shows Gaussian decay with a 1/e lifetimeof T*₂=4.5(1) μs, limited by thermal Doppler shifts. Inserting anadditional π-pulse 830 between the π/2-pulses 832, 834 cancels theeffect of the Doppler shifts and results in a substantially longercoherence lifetime of T₂=32(6) μs (fitted to an exponential decay downto 0.5). At 10 μK, the random atomic velocity in each shot of theexperiment appears as a random detuning δ^(D) from a Gaussiandistribution of width 2π×43.5 kHz, resulting in dephasing as

${ {  {❘\psi} \ranglearrow{\frac{1}{\sqrt{2}}{({❘g}}} \rangle + {e^{i\delta^{D}t}{❘r}}} \rangle)}.$

However, since the random Doppler shift is constant over the duration ofeach pulse sequence, its effect can be eliminated via a spin-echosequence (see plot 820 in FIG. 8B). The spin-echo measurements displaysome small deviations from the numerical simulations (dotted lines),indicating the presence of an additional dephasing channel. Assuming anexponential decay, the fitted T₂=32(6) μs and the pure dephasing timeT_(ϕ)=(1/T₂−1/(2T_(r→g)))⁻¹=47(13) μs. This dephasing may result fromresidual laser phase noise. Apart from resonantly manipulating the atomsbetween their states |g> and |r> it is also desirable to be able tomanipulate the phase between these states, which can be referred to as aphase gate. FIG. 8C shows such a single-atom phase gate implemented byapplying an independent 809 nm laser 840. According to some embodiments,other wavelengths that are far away from atomic transitions, butsufficiently close to induce a light shift on the ground state that aredifferent to that of the Rydberg state may be used. This induces a lightshift δ=2π×5 MHz on the ground state for time t, resulting in anaccumulated dynamical phase ϕ=δt. The gate may be embedded in aspin-echo sequence to cancel Doppler shifts. In each measurement shownhere, the 1013 nm laser remains on for the entire pulse sequence, whilethe 420 nm laser is pulsed according to the sequence shown above eachplot. Each data point is calculated from 200-500 repeated measurementson single atoms, with error bars denoting 68% confidence intervals. InFIGS. 8B and 8C, the solid lines are fits to experimental data. Dottedlines show the contrast expected from the numerical model, includingfinite detection fidelity. Such long coherence times and single qubitcontrol gates are useful for quantum computing.

According to some embodiments, a single-atom phase gate can beimplemented by applying an independent focused laser that shifts theenergy of the ground state |g

by 5 MHz. By controlling the duration of the applied laser pulse, acontrolled dynamical phase can be imparted on |g

relative to |r

. The contrast of the resulting phase gate (embedded in a spin-echosequence) is close to the limit imposed by detection and spin-echofidelity.

According to some embodiments, two-atoms may be controlled. It should beappreciated that such techniques and systems can be applied to more thantwo atoms. To this end, two atoms may be positioned at a separation of5.7 μm, at which the Rydberg-Rydberg interaction is U/ℏ=2π×30MHz >>Ω=2π×2 MHz. In this regime, which may be called a Rydberg blockaderegime, the laser field globally couples both atoms from |gg

to the symmetric state

$  { { {❘W} \rangle = {\frac{1}{\sqrt{2}}( {❘{gr}} }} \rangle \pm {❘{rg}}} \rangle )$

at an enhanced Rabi frequency of √{square root over (2)}Ω (see FIG. 9A).FIG. 9A shows the level structure for two nearby atoms, which features adoubly excited state |rr

which is shifted by the interaction energy U>>ℏΩ. In this Rydbergblockade regime, the laser field only couples |gg

to |W

. The symmetric and antisymmetric states |W

,

$  { { {❘D} \rangle = {\frac{1}{\sqrt{2}}( {❘{gr}} }} \rangle \pm {❘{rg}}} \rangle )$

can be coupled by a local phase gate on one atom (denoted via arrow910).

The probabilities can be measured for the states |gg

, |gr

, |rg

, and |rr

(denoted by P_(gg), P_(gr), P_(rg) and P_(rr), respectively), and showthat no population enters the doubly-excited state see FIG. 9B top(P_(rr)<0.02, consistent with only detection error). Instead, there areoscillations between the manifold of zero excitations, FIG. 9B bottom,and the manifold of one excitation, FIG. 9B center, with a fittedfrequency of 2π×2.83 MHz≈√{square root over (2)}Ω (see FIG. 9B). Afterdriving both atoms on resonance for variable time, the probability ofthe resulting two-atom states can be measured. Population oscillatesfrom |gg

to |W

at the enhanced Rabi frequency √{square root over (2)}Ω. Thisdemonstrates high fidelity two qubit control.

These collective Rabi oscillations can be used to directly prepare themaximally entangled Bell state |W

by applying air-pulse at the enhanced Rabi frequency (denoted by X_(π)^(W)). To determine the fidelity of this experimentally preparedentangled state, given by

=

W|ρ|W), it may be expressed in terms of diagonal and off-diagonal matrixelements of the density operator p:

=(ρ_(gr,gr)+ρ_(rg,rg))+½(ρ_(gr,rg)+ρ_(rg,gr))  (3)

where ρ_(αβ,γδ)=

αβ|ρ|γδ

for α, β, γ, δ∈{g, r}. The diagonal elements can be directly measured byapplying a π-pulse and then measuring the populations. The resultsclosely match those of a perfect |W

state after accounting for state detection errors, with ρ_(gr,gr)ρ_(rg,rg)=0.94(1), relative to a maximum possible value of 0.95(1).

To measure the off-diagonal elements of the density matrix, thesingle-atom phase gate Z_(ϕ) ⁽¹⁾ 920 may be used, as demonstrated inFIG. 9C, which introduces a variable phase on one atom. For example, alocal beam adds a light shift δ to |gr

but not to |rg

, such that

$  {  {❘W} \ranglearrow{\frac{1}{\sqrt{2}}( {e^{i\delta t}{❘{gr}}} } \rangle + {❘{rg}}} \rangle ).$

This phase accumulation rotates |W

into the orthogonal dark state

$  { { {❘D} \rangle = {\frac{1}{\sqrt{2}}( {❘{gr}} }} \rangle - {❘{rg}}} \rangle )$

according to:

|W

→cos(δt/2)|W

+i sin(δt/2)|D

  (4)

Since |D

is uncoupled by the laser field, a subsequent π-pulse maps only thepopulation of |W

back to |gg

. The probability of the system to end in |gg

therefore depends on the phase accumulation time as P_(gg)(t)=Acos²(δt/2). Here, the amplitude of the oscillation A precisely measuresthe off-diagonal matrix elements (ρ_(gr,rg)=ρ_(rg,gr)). In order tomitigate sensitivity to random Doppler shifts, this entire sequence maybe embedded in a spin-echo protocol (see FIG. 9C). FIG. 9C shows ameasurement of the entanglement fidelity of the two atoms after aresonant π-pulse in the blockade regime. A local phase gate Z_(ϕ) ⁽¹⁾rotates |W

into |D

, which is detected by a subsequent π-pulse. The fitted contrast 0.88(2)measures the off-diagonal density matrix elements. The phase gate isimplemented by an off-resonant laser focused onto one atom, with acrosstalk of <2%. The measurement is embedded in a spin-echo sequence tocancel dephasing from thermal Doppler shifts. As described above, thephase gate is a useful single qubit gate for quantum computation. Here,it may be used to be able to characterize the entanglement between thetwo atoms. The echo sequence used here helps to cancel noise fromDoppler shifts which thereby increases the coherence of the system.

The resulting contrast was A=0.88(2)=2ρ_(gr,rg)=2ρ_(rg,gr). Combiningthese values with the diagonal matrix elements, entanglement fidelity ofF=0.91(2) was measured. The maximum measurable fidelity given theexperimental state detection error rates would be 0.94(2), so aftercorrecting for imperfect detection, the entangled Bell state was foundto have been created with fidelity of F=0.97(3). This fidelity includeserrors introduced during the pulses that follow the initial π-pulse, andtherefore constitutes a lower bound on the true fidelity.

Entanglement is a useful resource in quantum computation. However,entangled states can be very fragile and subject to fast dephasing. Themethod discussed herein may be used to protect entangled states againstcertain noise sources. According to some embodiments, the lifetime ofthe entangled state by exciting |W

may be explored with a π-pulse and then de-exciting after a variabledelay (see FIG. 10). FIG. 10 shows extension of entangled-state lifetimevia dynamical decoupling. The lifetime of |W

can be measured by exciting |gg

to |W

and then de-exciting after a variable time as shown in the plot 810. Thelifetime is limited by dephasing from random Doppler shifts. Insertingan additional 2π-pulse 1030 in the blockade regime swaps the populationsof |gr

and |rg

to refocus the random phase accumulation, extending the lifetime to ˜36μs as shown in plot 1020 (fitted to an exponential decay, shown as thesolid). The initial offset in each curve 1010, 1020 is set by the groundstate detection fidelity associated with the given trap-off time. Alldata points are calculated from 30-100 repeated measurements, averagingover nine independent identically-coupled atom pairs, with error barsindicating 68% confidence intervals. Dotted lines near the plots and fitlines show predictions from the numerical model, including detectionerror. The decay in contrast is in good agreement with numericalpredictions associated with random Doppler shifts. In particular, thetwo components |gr

and |rg

of the |W

state dephase as

${ {  {❘W} \ranglearrow{\frac{1}{\sqrt{2}}( {e^{i\delta_{2}^{D}t}{❘{gr}}} } \rangle + {e^{i\delta_{1}^{D}t}{❘{rg}}}} \rangle)},$

where δ_(i) ^(D) is the two-photon Doppler shift on atom i.

According to some embodiments, the lifetime of the two-atom entangledstate can be extended with an echo sequence that acts on multiplequbits. This allows for longer periods of control. After the |W

state has evolved for time T, a 2π-pulse can be applied to the two-atomsystem. In the Rydberg blockade regime, such a pulse swaps thepopulations of |gr

and |rg

. After again evolving for time T, the total accumulated Doppler shiftsare the same for each part of the two-atom wavefunction, and thereforedo not affect the final |W

state fidelity. FIG. 10 shows that its lifetime is extended far beyondthe Doppler-limited decay to T₂ ^(W)=36(2) μs. As in the single atomcase, a pure dephasing timescale T_(ϕ) ^(W)=(1/T₂ ^(W)−1/T_(r→g))⁻¹>100μs is extracted.

The Bell state dephasing time T_(ϕ) ^(W)>100 μs of the two atoms issignificantly longer than the single atom dephasing time T_(Ø)=47(13)μs. This can be understood by noting that the states |gr

and |rg

form a decoherence-free subspace that is insensitive to globalperturbations such as laser phase and intensity fluctuations that coupleidentically to both atoms. In contrast, a single atom in a superposition

$  { { {❘\psi} \rangle = {\frac{1}{\sqrt{2}}( {❘g} }} \rangle + {❘r}} \rangle )$

is sensitive to both the laser phase and the laser intensity. Suchdecoherence free subspaces may be used to protect quantum informationfrom certain noise sources. These measurements provide furtherindications that even though the laser noise is significantly reduced inthese experiments, it is still not completely eliminated in ourexperiment A higher finesse cavity REF 720 may be used to filter outeven more laser noise and enable even longer coherence times.Additionally, these coherent manipulation techniques between the groundand Rydberg states are nonetheless significantly better than thosepreviously reported.

These measurements establish Rydberg atom qubits as a platform forhigh-fidelity quantum simulation and computation. The techniquesdemonstrated in this disclosure show methods of controlling a neutralatom arrays. The fidelities demonstrated by these techniques can befurther improved by increasing laser intensities and operating at largerdetunings from the intermediate state, thereby reducing the deleteriouseffect of off-resonant scattering, or by using a direct single-photontransition. In addition, sideband cooling of atoms in tweezers candramatically decrease the magnitude of Doppler shifts, while low-noiselaser sources such as Titanium-Sapphire lasers or diode lasers filteredby higher-finesse cavities can further eliminate errors caused by phasenoise. Advanced control techniques, such as laser pulse shaping, canalso be utilized to reach higher fidelities. Finally, state detectionfidelities, the major source of imperfections in the present work, canbe improved by field ionization of Rydberg atoms or by mapping Rydbergstates to separate ground state levels.

Section 5.A: Examples—Quantum Dynamics Across a Phase Transition

The system and methods described herein provide for identifications forsolutions of the Ising Problem, as discussed below. The techniquesapplied herein may also be transferred to other models, such as themaximum independent set problem described below.

Once atoms can be arranged in large arrays, such as a 1D array of asmany as 51 atoms or more, phase transitions can be observed as atomsalternate between the Rydberg and ground state. These transitions arediscussed in more detail below. FIGS. 5A-D shows characteristics of thetransition into the Z₂-phase in an array of 51 atoms, according to anembodiment. Long ordered chains where the atomic states alternatebetween Rydberg and ground state may appear. As shown in FIG. 5A, theseordered domains can be separated by domain walls that consist of twoneighboring atoms in the same electronic state.

FIG. 5A shows single-shot fluorescence images of a 51-atom array beforeapplying the adiabatic pulse (top row 501, i.e., the evolution step 116as discussed with respect to FIG. 1D) and after the pulse (bottom threerows 502, 503, 504 correspond to three separate instances, i.e., thedetection step 118 in FIG. 1D), according to an exemplary embodiment.Small circles 505 mark lost atoms, which may be attributed to Rydbergexcitations. Domain walls are defects in the perfectly ordered Rydbergcrystal. Domain walls allow for characterization of how well the systemreaches the ground state at the end of an adiabatic sweep. Observingthese domain walls themselves are useful as well. For example, thesystem can be better characterized by how the sweep speed influences thenumber of domain walls or whether there are correlations between domainwalls. Domain walls 506 (circled dots) are identified as either twoneighboring atoms in the same state or a ground state atom at the edgeof the array and are indicated with an ellipse. Long Z₂ ordered chainsbetween domain walls can be observed.

FIG. 5B shows domain wall density as a function of detuning during thefrequency sweep, according to an exemplary embodiment. The points oncurve 560 are the mean of the domain wall density as a function ofdetuning during the sweep. Error bars are standard error of the mean andare smaller than the marker size. The points on curve 570 are thecorresponding variances where the shaded region represents the jackknifeerror estimate. The onset of the phase transition is witnessed by adecrease in the domain wall density and a peak in the variance. Eachpoint is obtained from 1000 realizations. The solid curve 560 is a fullycoherent MPS simulation without free parameters (bond dimension D=256),taking measurement fidelities into account.

The domain wall density can be used to quantify the transition from thedisordered phase into the ordered Z₂-phase as a function of detuning Δand serves as an order parameter. As the system enters the Z₂-phase,ordered domains grow in size, leading to a substantial reduction in thedomain wall density (points on curve 560 in FIG. 5B). Consistent withexpectations for an Ising-type second order quantum phase transition,domains of fluctuating lengths close to the transition point between thetwo phases can be observed, which is reflected by a pronounced peak inthe variance of the density of domain walls. This peak is shiftedtowards positive values of Δ≈0.5Ω, consistent with predictions fromfinite size scaling analysis. The observed domain wall density is inexcellent agreement with fully coherent simulations of the quantumdynamics based on 51-atom matrix product states (line 560); however,these simulations underestimate the variance at phase transition.

At the end of the sweep, deep in the Z₂ phase (Δ/Ω>>1), Ω can beneglected such that the Hamiltonian (1) becomes essentially classical.In this regime, the measured domain wall number distribution allows usto directly infer the statistics of excitations created when crossingthe phase transition. FIG. 5C shows a histogram of the normalized numberof domain walls that appeared during 18439 separate experimentalrealizations of the 51-atom array. The distribution is depicted in withan average of 9.01(2) domain walls. This distribution is affected by thedetection fidelity of state |g

and state |r

which leads to a higher number of domain walls. In other words, a systemwith perfect detection fidelity would produce a different distributionof domain walls than the one employed in which imperfect detectionfidelity introduces additional domain walls. Thus, to determine the realnumber of domain walls the effect of detection fidelity can be modeledon the domain wall distribution to determine the number of domain wallsthat were created without the effects of lower detection fidelity. Thisallows a maximum likelihood estimation to be obtained the distributioncorrected for detection fidelity, which corresponds to a state that hason average 5.4 domain walls. These remaining domain walls (those notcaused by lower detection fidelity) are most likely created due tonon-adiabatic transitions from the ground state when crossing the phasetransition, where the energy gap becomes minimal. In addition, thepreparation fidelity is also limited by spontaneous emission during thelaser pulse (an average number of 1.1 photons is scattered per μs forthe entire array).

FIG. 5C shows domain wall number distribution for A=14 MHz obtained from18439 experimental realizations (top plot), according to an exemplaryembodiment. Error bars indicate 68% CI. Owing to the boundaryconditions, only even number of domain walls can appear. Bars on theright of each coupled pair in the bottom plot show the distributionobtained by correcting for finite detection fidelity using a maximumlikelihood method, which results in an average number of 5.4 domainwalls. Bars to the left of each coupled pair show the distribution of athermal state with the same mean domain wall density. FIG. 5D sowsmeasured correlation function in the Z2 phase.

To further characterize the created Z₂ ordered state, the correlationfunction can be evaluated

g _(ij) ⁽²⁾ =

n _(i) n _(j)

−

n _(i)

n _(j)

  (2)

where the average < . . . > is taken over experimental repetitions. Wefind that the correlations decay exponentially over distance with adecay length of ξ=3.03(6) sites (see FIG. 5d and SI).

FIG. 6 shows graphical demonstrates that the approach described withinthe present disclosure also enables the study of coherent dynamics ofmany-body systems far from equilibrium. FIG. 6A shows a schematicsequence (top, showing A(0) involves adiabatic preparation and then asudden quench to single-atom resonance. The heat map shows the singleatom trajectories for a 9-atom cluster. The initial (left inset) crystalis observed with a Rydberg excitation at every odd trap site collapsesafter the quench and a crystal with an excitation at every even sitebuilds up (middle inset). At a later time, the initial crystal revives(right inset). Error bars denote 68% CI. FIG. 6B shows the density ofdomain walls after the quench. The dynamics decay slowly on a timescaleof 0.88 μs. The shaded region represents the standard error of the mean.Solid line in the top pane is a fully coherent MPS simulation with bonddimension D=256, taking into account measurement fidelity. FIG. 6C showsa toy model of non-interacting dimers. FIG. 6D shows numericalcalculations of the dynamics after a quench starting from an ideal 25atom crystal, obtained from exact diagonalization. Domain wall densityas a function of time 610, and growth of entanglement entropy of thehalf chain (13 atoms) 620. Dashed lines take into account only nearestneighbor blockade constraint. Solid lines correspond to the full 1/r⁶interaction potential.

As shown in FIG. 6A, focusing on the quench dynamics of Rydberg crystalsinitially prepared deep in the Z₂ ordered phase, the detuning Δ(t) issuddenly changed to the single-atom resonance Δ=0. After such a quench,oscillations of many-body states appear between the initial crystal anda complementary crystal where each internal atomic state is inverted.These oscillations are remarkably robust, persisting over severalperiods with a frequency that is largely independent of the system sizefor large arrays. This is confirmed by measuring the dynamics of thedomain wall density, signaling the appearance and disappearance of thecrystalline states, shown in FIG. 6B for arrays of 9 (solid dots) and 51atoms (open dots). The initial crystal repeatedly revives with a periodthat is slower by a factor 1.4 compared to the Rabi oscillation periodfor independent, non-interacting atoms.

According to an embodiment, several important features result from thetechniques described in the present disclosure. First, the Z₂ orderedstate cannot be characterized by a simple thermal ensemble. Morespecifically, if an effective temperature is estimated based on themeasured domain wall density, the corresponding thermal ensemblepredicts correlation length ξth=4.48(3), which is significantly longerthan the measured value ξ=3.03(6). Such a discrepancy is also reflectedin distinct probability distributions for the number of domain walls(see FIG. 5c ). These observations suggest that the system does notthermalize within the timescale of the Z₂ state preparation

Second, the coherent and persistent oscillation of the crystalline orderafter the quantum quench. With respect to the quenched Hamiltonian(Δ=0), the energy density of the Z₂ ordered state corresponds to that ofan infinite temperature ensemble within the manifold constrained byRydberg blockade. Also, the Hamiltonian does not have any explicitconserved quantities other than total energy. Nevertheless, theoscillations persist well beyond the natural timescale of localrelaxation ˜1/Ω as well as the fastest timescale,

To understand these observations, consider a simplified model where theeffect of long-range interactions are neglected and nearest-neighborinteractions are replaced by hard constraints on neighboring excitationsof Rydberg states. In this limit, the qualitative behavior of thequenched dynamics can be understood in terms of dimerized spins (asshown in FIG. 6C); owing to the constraint, each dimer forms aneffective spin-1 system with three states |r,g

, |gg

, and |gr

where the resonant drive “rotates” the three states over the period2(2π/Ω), close to that observed experimentally. While this qualitativepicture does not take into account the strong interactions (constraints)between neighboring dimers, it can be extended by considering a minimalvariational ansatz for the many-body wave function based on matrixproduct state (MPS) that respects all Rydberg blockade constraints.Using time-dependent variational principle, analytical equations ofmotion may be derived and may obtain a crystalline order oscillationwith frequency Ω/1.51, which is within 10% of the experimentalobservations. These considerations may be supported by various numericalsimulations. For example, an MPS simulation with large bond dimensionpredicts that the simplified model exhibits crystal oscillation overlong time, while the entanglement entropy grows at a rate much smallerthan Ω, indicating that the oscillation persists over many cycles (FIG.6D). However, the addition of long-range interactions leads to a fasterdecay of the oscillations, with a timescale that is determined by1/V_(i,i+2), consistent with experimental observations, while theentanglement entropy also grows on this time scale.

Thus, the decay of crystal oscillation is limited by the effect of weaknext-nearest-neighbor interactions. This slow thermalization is ratherunexpected since our Hamiltonian, with or without long-rangecorrections, is far from any known integrable systems and featuresneither strong disorder nor explicitly conserved quantities. Instead,observations may be associated with constrained dynamics due to Rydbergblockade, resulting in large separations of timescales V_(i,i+1)»ΩV_(i,i+2). These give rise to so-called constrained dimer models, withHilbert space dimension determined by Golden ratio (1+5)^(N)/2^(N) andnon-trivial dynamics.

According to an embodiment, the initial spacing between the atomstrapped in an array may be used in order to encode a problem, such asone governed by quantum mechanics. After adiabatic evolution of thesystem, the atoms may then be observed to determine a solution to theproblem. The state of the atoms after evolution may be indicative of asolution to the problem.

Section 5.B: Examples—Solving Maximum Independent Set OptimizationProblems Using Quantum Computers

The methods and systems described above for arranging and controllingindividually trapped neutral atoms and their Rydberg inter-actions maybe used to solve a variety of different types of problems. For example,as described below, according to some embodiments, the systems andmethods described above can be used to solve maximum independent set(MIS) optimization problems based on the quantum adiabatic principle.MIS optimization problems are challenging to solve using numericaltechniques alone but can be more easily solved using quantum computingtechniques. Thus, the above described systems and methods for quantumcomputing are well suited for finding solutions to the MIS optimizationproblems described below.

Adiabatic quantum computation is a new, general approach to solvingcombinatorial optimization problems. It consists of constructing a setof qubits and engineer a time-dependent Hamiltonian H(t) whose startingpoint H(0) has a ground state that can be easily prepared and whosefinal point H(T) has such a form that its eigenstates encode thesolution to the optimization problem. The name “adiabatic” originatesfrom the fact if H(t) is varied slowly enough, then the system willmostly stay in the ground state of the instantaneous Hamiltonian H(t) atall times t, such that at the final time t=T the system is found in theground state of H(T), from which the solution to the optimizationproblem can be found. According to some embodiments, if the Hamiltonianis not changed slowly enough for a fully adiabatic evolution, thedynamics induced by the time dependent Hamiltonian may inject a finiteenergy into the system. As long as the cost function of the optimizationproblem is correctly encoded in the final Hamiltonian, and the evolutionis slow enough that the injected energy is low, measuring the finalstate of the system gives a good approximate solution of theoptimization problem. The working principle of this quantum adiabaticoptimization (or approximation) is fundamentally different from knownclassical algorithms to find (or approximate) solutions to optimizationproblems and thus can lead to a quantum speedup (i.e., performingcomputations faster using quantum computers).

The maximum independent set problem (as discussed in more detail below)is a classical combinatorial optimization problem in graph theory. Thetask is to select a subset of vertices from a graph, such that none ofthem are not neighboring. In some embodiments, the challenge is to findsuch a subset that with the largest number of vertices. It is awell-studied problem in complexity theory and it is known to be NP-hardto approximate. When formulated as decision problem it is NP-complete(i.e., belonging both to NP (nondeterministic polynomial time) problemsand NP-hard (problems that are at least as hard as NP problems)problems).

According to some embodiments, there are different variants of themaximum independent set problem. The embodiments discussed herein focuson problems where the class of graphs can be restricted to disc graphsfor which optimization of the maximum independent set problem isdesired. Unit disc graphs are a special case of geometric intersectiongraphs (i.e. graphs that represents the pattern of intersections of afamily of sets), where only vertices that are located within a certaindistance are considered neighboring. Optimization problems on suchintersection graphs play an important role in a variety of applications,including, but not limited to problems arising in broadcast networkdesign, map labelling, and determination of the optimal location forfacilities. The maximum independent set problem on unit disk graphs isNP-complete. While but polynomial time approximation algorithms exist,fast algorithms that achieve good approximation ratios has not yet beenachieved.

As discussed in more detail below a setup including individually trappedatoms can be used to implement quantum algorithms to solve the maximumindependent set problem on unit disc graphs.

As described herein, the maximum independent set problem may include anundirected graph G=(V, E) with a set of vertices V and edges E. Anindependent set is a subset of vertices S⊆V such that no two vertices inS are connected by an edge. For examples of independent sets see FIGS.11A-11B. FIG. 11A show two examples of graphs with different independentsets marked as combinations of empty and black circles. The maximumindependent set in each case is depicted on the right. In other words,the maximum independent set is the largest independent set, i.e. theindependent set with the largest number of vertices. Note: there is alsothe notion of a maximal independent set, which is an independent set towhich one cannot add any other vertex without making it not independent.The maximum independent set is the largest of the maximal independentsets.

A generalization of the maximum independent set problem is themaximum-weight independent set problem. According to some embodiments,an undirected weighted graph G=(V, W, E) may be considered with a set ofvertices V with associated weights W and edges E. The maximum-weightindependent set is the independent set with the largest weight. Themaximum independent set problem can be formulated as a decision problem:“Given a graph G, is there an independent set of size k?”. This decisionproblem is NP-complete. It can also be formulated as an optimizationproblem: “Given a graph G, find the maximum size k of independent sets”.Even approximate optimization is NP-hard (approximation of k, within aconstant factor). Finding the maximum independent set is equivalent tofinding the minimum vertex cover: these are dual problems. Theseprinciples extend to the maximum-weight independent set problems.

A graph may be called a unit disc graph if only vertices that are withina unit distance R in an Euclidian space are connected by an edge. FIG.12 shows an example of a unit disc graph. As shown in FIG. 2, every pairof vertices 1210 that is closer that a distance R is connected by anedge 1220. This is equivalent to drawing discs 1230 of radius R/2 aroundeach vertex and connecting them if two discs overlap. Note that findingthe maximum independent set of unit disc graphs is still NP-complete.

According to some embodiments, arrays of atoms may be arranged in orderto solve problems such as those presented by unit disc graphs. Accordingto some embodiments, Rydberg atoms may be used, in which the Rydbergimplementation R plays the role of the blockade radius, whichcorresponds to the discs 1230. Such implementations are discussed inmore detail below.

Given a graph, the maximum independent set can be found from the groundstate of a classical Hamiltonian. To this end a classical Ising variableto may be assigned to each vertex, n_(v)−{0, 1}. The Hamiltonian isshown below

$\begin{matrix}{H = {{\sum\limits_{v \in V}{{- \Delta}n_{v}}} + {\sum\limits_{{({u,w})} \in E}{U_{u,w}n_{u}n_{w}}}}} & (5)\end{matrix}$

with U_(u,w)>Δ>0. The configuration {n_(v)} that minimizes H encodes themaximum independent set: all vertices that in the ground state haven_(v)=1 form the maximum independent set. Note that the value of U_(n,m)is not important as long as it is larger than Δ.

Moreover note that (for U_(n,m)>>Δ) the lowest lying energy statesencode different independent sets, and note further that thecorresponding energy is directly related to the size of the independentset k as E=−kΔ.

To design a quantum adiabatic algorithm for MIS, the Hamiltonian can bepromoted to an operator level including the addition of a term thatcouples different configurations of the Ising spin. For example, theHamiltonian may be written as

$\begin{matrix}{{\overset{\hat{}}{H}(t)} = {{\sum\limits_{v \in V}( {{{- {\Delta(t)}}{\overset{\hat{}}{n}}_{v}} + {{\Omega(t)}{\overset{\hat{}}{\sigma}}_{v}^{x}}} )} + {\sum\limits_{{({u,w})} \in E}{U_{u,w}{\overset{\hat{}}{n}}_{u}{\overset{\hat{}}{n}}_{w}}}}} & (6)\end{matrix}$

Instead of classical Ising spins, there are now qubits with states |0

and |1

such that {circumflex over (n)}|x

=x|x

, (x∈{0,1}), and σ^(x)=|0

1|+|1

0|. An adiabatic algorithm can be thus obtained by initializing allqubits at time t=0 in |0

and then evolving the system under the time dependent Hamiltonian H(t)for a time T with parameters chosen such that Δ(0)<0, Δ(T) >0,Ω(0)=Ω(T)=0 and Ω(0<t<T) >0. As a specific example consider Δ(t)=(2t/T−1){tilde over (Δ)} with {tilde over (Δ)}>0, Ω(t)={tilde over (Ω)}sin²(πt/T). Note that this is immediately generalized to themaximum-weight independent set problem by making the parameter Δ(t)different for each vertex.

According to some embodiments, the system and method of arranging andmanipulating individual atoms described in more detail above may be usedto encode and evolve such problems. For example, a set of individuallypositioned optical tweezers as discussed in more detail above may beused to each trap a single atom with a ground state |0

and Rydberg state |1

. The atoms can be coherently driven with Rabi frequency Ω(t) couplingthe ground state to the Rydberg state. The frequency of the drivingfield can be changed time dependently, giving rise to a time dependentdetuning Δ(t). This driving can be either global, or alternatively eachatom can be individually driven with a particular field at particulartimes. If two atoms u and v are in a Rydberg state they interact,shifting the energy of this configuration by an amount W_(u,v) whichdepends on the geometric distance d_(u,v)=|{right arrow over(x)}_(u)−{right arrow over (x)}_(v)| between the two trap locations,e.g. W_(u,v)=C/d_(u,v) ⁶. The Hamiltonian describing the dynamics ofthis array for trapped atoms is thus:

$\begin{matrix}{{\overset{\hat{}}{H}(t)} = {{\sum\limits_{v \in V}( {{{- {\Delta(t)}}{\overset{\hat{}}{n}}_{v}} + {{\Omega(t)}{\overset{\hat{}}{\sigma}}_{v}^{x}}} )} + {\sum\limits_{u,{w \in V}}{W_{u,w}{\overset{\hat{}}{n}}_{u}{\overset{\hat{}}{n}}_{w}}}}} & (7)\end{matrix}$

For two atoms that are trapped in close proximity it is energeticallyextremely costly to simultaneously populate the Rydberg state.

Since the unit disc graph has a geometric interpretation, the traps maybe arranged according to the arrangement of the vertices in the unitdisc graph. The unit of length is chosen such that the Rydberg blockaderadius corresponds to the unit distance in the graph, that is such that

W _(u,v)>Δ(T), if d _(u,v) <R  (8)

W _(u,v)<Δ(T), if d _(u,v) >R  (9)

The quantum optimization algorithm can be implemented experimentally byslowly changing the parameters Ω(t) and Δ(t), and measuring at the endwhich atoms are in the Rydberg state. If the evolution is slow enough,this will be he maximum independent set. If the evolution is notperfectly adiabatic but the injected energy is low, the final state willin general be a superposition of “independent set states”, that isconfigurations with atoms in the Rydberg state if they are not withinthe Blockade radius. The larger the time T, the better the approximationratio that the protocol can achieve will be.

The above-described method of encoding MIS problems neglectsinteractions that are smaller than the blockade interaction. If the longtail interactions are included, a geometric arrangement of the traps maybe chosen such that all traps v are defined by 0<Δ+δ_(v)<W_(v,w)∀w|(v,w)∈E, where δ_(v)=Σ_(u|(u,v)∉E)W_(v,u) is the largest possibleenergy shift that can arise for an atom in the Rydberg state at vertex vdue to interactions outside the Blockade radius. Thus as long as δ_(v)is small (i.e. interactions between atoms outside the blockade Radiuscan be neglected), the quantum algorithm gives (or approximates)solutions to the maximum independent set problem.

FIG. 13A shows an example of a unit disc graph and indicates the maximumindependent set. As shown in FIG. 13A, a unit disc graph has 25 vertices(small circles 1310) and vertex density of 2.7. The vertices 1320 at thecenter of the larger circles 1330 consist of a maximum independent set(there is more than one). The larger circles 1330 indicate the blockaderadius. FIG. 13B shows the probability distribution of finding anindependent set of size k when the quantum algorithm is run for a time TThe longer time T, the higher the probability that the algorithm revealsa large (or even the maximum) independent set. The probabilitydistribution of the size of the independent set was found by theadiabatic algorithm after an evolution under Hamiltonian 2 with a timeT. Here units are such that {tilde over (Δ)}=7.5 and {tilde over (Ω)}=1.Already for a time T˜5 the probability to find the global optimum issubstantial.

The techniques described in the present disclosure can also include anumber of variations or applications. For example, coherence propertiesof atoms can be improved by increasing intermediate state detuning tofurther suppress spontaneous emission and by Raman sideband coolingatomic motion to the ground state to eliminate the residual Dopplershifts. Individual qubit rotations around the z-axis can be implementedusing light shifts associated with trap light, while a second AOD can beused for individual control of coherent rotations around otherdirections. Further improvement in coherence and controllability can beobtained by encoding qubits into hyperfine sublevels of the electronicground state and using state-selective Rydberg excitation. Implementingtwo-dimensional (2d) may be implemented to make thousands of traps. Such2d configurations may be implemented by directly using a 2d-AOD or bycreating a static 2d lattice of traps and sorting atoms with anindependent AOD.

It is to be understood that the disclosed subject matter is not limitedin its application to the details of construction and to thearrangements of the components set forth in the following description orillustrated in the drawings. The disclosed subject matter is capable ofother embodiments and of being practiced and carried out in variousways. Also, it is to be understood that the phraseology and terminologyemployed herein are for the purpose of description and should not beregarded as limiting.

As such, those skilled in the art will appreciate that the conception,upon which this disclosure is based, may readily be utilized as a basisfor the designing of other structures, methods, and systems for carryingout the several purposes of the disclosed subject matter. It isimportant, therefore, that the claims be regarded as including suchequivalent constructions insofar as they do not depart from the spiritand scope of the disclosed subject matter.

Although the disclosed subject matter has been described and illustratedin the foregoing exemplary embodiments, it is understood that thepresent disclosure has been made only by way of example, and thatnumerous changes in the details of implementation of the disclosedsubject matter may be made without departing from the spirit and scopeof the disclosed subject matter, which is limited only by the claimswhich follow.

The techniques and systems disclosed herein, such as particular AODs orlaser systems, may controlled, for example, by using a computer programproduct for use with a network, computer system or computerizedelectronic device. Such implementations may include a series of computerinstructions, or logic, fixed either on a tangible medium, such as acomputer readable medium (e.g., a diskette, CD-ROM, ROM, flash memory orother memory or fixed disk) or transmittable to a network, computersystem or a device, via a modem or other interface device, such as acommunications adapter connected to a network over a medium.

The medium may be either a tangible medium (e.g., optical or analogcommunications lines) or a medium implemented with wireless techniques(e.g., Wi-Fi, cellular, microwave, infrared or other transmissiontechniques). The series of computer instructions embodies at least partof the functionality described herein with respect to the system. Thoseskilled in the art should appreciate that such computer instructions canbe written in a number of programming languages for use with manycomputer architectures or operating systems.

Furthermore, such instructions may be stored in any tangible memorydevice, such as semiconductor, magnetic, optical or other memorydevices, and may be transmitted using any communications technology,such as optical, infrared, microwave, or other transmissiontechnologies.

It is expected that such a computer program product may be distributedas a removable medium with accompanying printed or electronicdocumentation (e.g., shrink wrapped software), preloaded with a computersystem (e.g., on system ROM or fixed disk), or distributed from a serveror electronic bulletin board over the network (e.g., the Internet orWorld Wide Web). Of course, some embodiments of the invention may beimplemented as a combination of both software (e.g., a computer programproduct) and hardware. Still other embodiments of the invention areimplemented as entirely hardware, or entirely software (e.g., a computerprogram product).

In the foregoing description, certain steps or processes can beperformed on particular servers or as part of a particular engine. Thesedescriptions are merely illustrative, as the specific steps can beperformed on various hardware devices, including, but not limited to,server systems and/or mobile devices. Alternatively or in addition, anyor all of the steps described herein can be performed on a virtualizedmachine that runs on a physical server itself. Similarly, the divisionof where the particular steps are performed can vary, it beingunderstood that no division or a different division is within the scopeof the invention. Moreover, the use of “module” and/or other terms usedto describe computer system processing is intended to be interchangeableand to represent logic or circuitry in which the functionality can beexecuted.

1. A system for configuring an array of atoms for performing quantumcomputation, comprising: an optical confinement system for arrangingatoms in a first array state in a trap array, the optical confinementsystem comprising: a holding trap array configured to hold a pluralityof atoms, said holding trap array having at least three traps; a firstcontrol acousto-optical deflector (AOD) and a second control AOD,wherein the first control AOD is configured to deflect the first laserbeam in a first direction and the second control AOD is configured todeflect the first laser beam in a second direction different from thefirst direction; an adjustable acoustic tone source configured toselectively apply a first plurality and a second plurality of acoustictones to the first and the second control AODs, respectively, each tonehaving a discrete adjustable frequency, each frequency corresponding toat least one trap of the holding trap array, and a first laser lightsource arranged to pass a beam through the first and the second controlAODs; a magneto-optical trap, the magneto-optical trap being configuredto position an atom cloud to at least partially overlap with theplurality of traps; a second laser light source for evolving at leastsome of the plurality of atoms in the trap array in the first arraystate into a second array state; and an imaging device for observing theplurality of atoms in the second array state.
 2. The system of claim 1,wherein the second laser light source is configured to excite the atleast some of the plurality of atoms in the first array state into aRydberg state.
 3. The system of claim 1, wherein the second laser lightsource is configured to excite the at least some of the plurality ofatoms into a Zeeman sublevel of their ground state.
 4. The system ofclaim 3, the second laser light source further comprises an opticalpumping system and a magnetic field generator.
 5. The system of claim 1,wherein the second laser light source is configured to generate photonshaving a first wavelength and a second wavelength, thereby producing atwo photon transition of the at least some of the plurality of atoms inthe first array state, wherein the first wavelength is approximately 420nm and the second wavelength is approximately 1013 nm.
 6. The system ofclaim 5, wherein the second laser light comprises a source having athird wavelength, the source configured to apply a phase gate bydirecting photons to the at least some of the trapped atoms, wherein thethird wavelength is approximately 809 nm.
 7. The system of claim 6,wherein the second laser light source is configured to apply two half-pipulses.
 8. The system of claim 7, wherein the second laser light sourceis configured to apply a pi pulse between the two half-pi pulses.
 9. Thesystem of claim 1, wherein the optical confinement system comprises atleast one of: at least one holding AOD, a spatial light modulator (SLM),or an optical lattice.
 10. A method of configuring an array of atoms forperforming quantum computation, comprising: forming an array of atoms ina first array state, wherein said forming comprises: trapping aplurality of atoms in a holding trap array, said holding trap arrayhaving at least three traps; passing a first laser beam through a firstcontrol acousto-optical deflector (AOD) and a second control AOD,wherein the first control AOD is configured to deflect the first laserbeam in a first direction and the second control AOD is configured todeflect the first laser beam in a second direction different from thefirst direction; exciting the first and the second control AODs with afirst plurality and a second plurality, respectively, of acoustic tones,each tone having a discrete adjustable frequency, each frequencycorresponding to at least one trap of the holding trap array; adjustingthe frequency of at least one tone of the first or second pluralities oftones, thereby moving at least two of the plurality of atoms in theholding trap array.
 11. The method of claim 10, wherein the holding traparray is formed by an element selected from: at least one holding AOD, aspatial light modulator (SLM), or an optical lattice.
 12. The method ofclaim 11, wherein trapping the plurality of atoms in the holding traparray comprises: exciting one of the at least one holding AOD with aplurality of acoustic tones, each tone having a discrete frequency, eachtrap of the holding trap array corresponding to the frequency of one ofthe plurality of acoustic tones; passing a second laser beam through theat least one holding AOD to create a plurality of traps; and trapping atleast two atoms in at least two of said plurality of traps.
 13. Themethod of claim 10, further comprising: causing a transition of at leastsome of the trapped atoms into an excited state by directing photonsthereto, thereby evolving the array of atoms from the first array stateinto a second array state; and observing the plurality of atoms in thesecond array state.
 14. The method of claim 10, wherein the firstdirection and the second direction have a relative angle of 60 degreestherebetween.
 15. The method of claim 10, wherein the first directionand the second direction have a relative angle of 90 degreestherebetween.
 16. The method of claim 10, wherein the array of atoms inthe first array state comprises between 7 and 51 atoms.
 17. The methodof claim 13, wherein evolving the plurality of atoms further comprisespreparing the at least some of the atoms into a Zeeman sublevel of theirground state.
 18. The method of claim 17, wherein preparing the at leastsome of the atoms into the Zeeman sublevel of the ground state comprisesoptical pumping in a magnetic field.
 19. The method of claim 13, whereinthe photons have wavelengths of approximately 420 nm and approximately1013 nm, and wherein the transition of the at least some of the atomsinto the excited state comprises a two photon transition.
 20. The methodof claim 19, further comprising applying a phase gate by directingphotons to the at least some of the trapped atoms, the photons having athird wavelength, wherein the third wavelength is approximately 809 nm.21. The method of claim 13, wherein directing the photons comprisesapplying two half-pi pulses.
 22. The method of claim 21, whereindirecting the photons further comprises applying a pi pulse between thetwo half-pi pulses.
 23. The method of claim 13, further comprising:preparing the array of atoms to execute a quantum computing problem;producing a solution to the quantum computing problem; and reading outthe quantum computing problem solution, wherein: preparing the array ofatoms to execute the quantum computing problem comprises moving the atleast two atoms in the holding trap array; producing the solution to thequantum computing problem comprises directing photons to the at leastsome of the trapped atoms to cause the transition of said atoms into theexcited state; and reading out the solution to the quantum computingproblem comprises observing the plurality of atoms in the second arraystate.
 24. The method of claim 23, wherein the quantum computing problemcomprises at least one of an Ising-problem and a maximum independent set(MIS) optimization problem.
 25. The method of claim 10, furthercomprising exciting the at least some of the plurality of atoms in thefirst array state into a Rydberg state.